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Impact Behaviors of Cantilevered Nano-beams Based on the Nonlocal Theory

  • C. Li
  • N. Zhang
  • X. L. Fan
  • J. W. YanEmail author
  • L. Q. Yao
Original Paper
  • 15 Downloads

Abstract

Purpose

The present study is motivated by the shock response of a nano-cantilever component in nano-scaled electromechanical system (NEMS). For this purpose, the mechanical properties of a cantilevered nano-beams subjected to impact load are presented for the first time. Our work is concerned with the special moment that the impact deformation gets most and the impact velocity reduces to zero.

Methods

The nonlocal theory proposed by Eringen is applied to reveal the nonlocal effect involved in impact behaviors at nano-scale. To this end, the classical impact stress is determined first using the law of conservation of energy. Subsequently, the nonlocal impact stress is obtained from the nonlocal constitutive equation and clamped-free boundary constraints of cantilevered nano-beams, where an imaginary equivalent symmetrical nano-structure is constructed to solve the unknown coefficients in general solution of stress of nonlocal field. The bending moment in nonlocal theory is gained from the stress of nonlocal field and the deflection in nonlocal field under impact load is calculated. Finally, the nonlocal dynamical load coefficient is derived via the maximal nonlocal impact deflection divided by the corresponding traditional static deflection at free edge. In numerical examples, the maximal stress of nonlocal field, the maximal nonlocal impact deflection, and the nonlocal dynamical load coefficient are calculated, respectively, to show the significant nonlocal small-scale effect in cantilevered nano-beams subjected to impact load.

Results and Conclusions

The conclusion is that the nonlocality plays a remarkable role in nano-scaled impact behaviors and it cannot be neglected. The existence of scale coefficient decreases the nonlocal impact stress, deflection, and dynamical load coefficient. On the contrary, the impact velocity results in a higher nonlocal impact stress, deflection, and dynamical load coefficient. With increasing the scale coefficient, the nonlocal effect increases and the strengthening physical manifestation of nano-structural equivalent stiffness becomes even more pronounced. The results reported herein are expected to provide a useful reference for understanding the scale effect in dynamic impact that often occurs in the working process of NEMS.

Keywords

Nano-beam Cantilever Nonlocal theory Dynamical load coefficient 

Notes

Acknowledgements

This work was supported by State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant No. MCMS-0418G01), the National Natural Science Foundation of China (Nos. 11972240, 11572210), and the Soochow Scholar Plan of Soochow University (No. R513300116).

References

  1. 1.
    Cai GP, Hong JZ, Yang SX (2005) Dynamic analysis of a flexible hub-beam system with tip mass. Mech Res Commun 32(2):173–190CrossRefzbMATHGoogle Scholar
  2. 2.
    Mao XY, Ding H, Chen LQ (2017) Vibration of flexible structures under nonlinear boundary conditions. J Appl Mech 84(11):111006CrossRefGoogle Scholar
  3. 3.
    Ding H, Dowell EH, Chen LQ (2018) Transmissibility of bending vibration of an elastic beam. J Vibr Acoust 140(3):031007CrossRefGoogle Scholar
  4. 4.
    Zhao J, Jiang JW, Jia Y, Guo W, Rabczuk T (2013) A theoretical analysis of cohesive energy between carbon nanotubes. Carbon 57:108–119CrossRefGoogle Scholar
  5. 5.
    Liu RM, Wang LF, Jiang JN (2016) Thermal vibration of a single-layered graphene with initial stress predicted by semiquantum molecular dynamics. Mater Res Exp 3(9):095601CrossRefGoogle Scholar
  6. 6.
    Skaug MJ, Schwemmer C, Fringes S, Rawlings CD, Knoll AW (2018) Nanofluidic rocking brownian motors. Science 359(6383):1505–1508CrossRefGoogle Scholar
  7. 7.
    Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710CrossRefGoogle Scholar
  9. 9.
    Li C, Lim CW, Yu JL (2011) Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Mater Struct 20(1):015023CrossRefGoogle Scholar
  10. 10.
    Şimşek M, Yurtcu HH (2013) Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct 97:378–386CrossRefGoogle Scholar
  11. 11.
    Li C, Guo L, Shen JP, He YP, Ju H (2013) Forced vibration analysis on nanoscale beams accounting for effective nonlocal size effects. Adv Vibr Eng 12(6):623–633Google Scholar
  12. 12.
    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 77:55–70MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li C (2014) A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries. Compos Struct 118:607–621CrossRefGoogle Scholar
  14. 14.
    Lim CW, Islam MZ, Zhang G (2015) A nonlocal finite element method for torsional statics and dynamics of circular nanostructures. Int J Mech Sci 94–95:232–243CrossRefGoogle Scholar
  15. 15.
    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 78:298–313MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li C, Yu YM, Fan XL, Li S (2015) Dynamical characteristics of axially accelerating weak visco-elastic nanoscale beams based on a modified nonlocal continuum theory. J Vibr Eng Technol 3(5):565–574MathSciNetGoogle Scholar
  17. 17.
    Ansari R, Oskouie MF, Gholami R, Sadeghi F (2016) Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Compos Part B Eng 89:316–327CrossRefGoogle Scholar
  18. 18.
    Yu YM, Lim CW (2013) Nonlinear constitutive model for axisymmetric bending of annular graphene-like nanoplate with gradient elasticity enhancement effects. J Eng Mech 139:1025–1035CrossRefGoogle Scholar
  19. 19.
    Li C (2016) On vibration responses of axially travelling carbon nanotubes considering nonlocal weakening effect. J Vibr Eng Technol 4(2):175–181Google Scholar
  20. 20.
    Xu XJ, Zheng ML, Wang XC (2017) On vibrations of nonlocal rods: boundary conditions, exact solutions and their asymptotics. Int J Eng Sci 119:217–231MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhu XW, Li L (2017) Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. Int J Mech Sci 133:639–650CrossRefGoogle Scholar
  22. 22.
    Yang Q, Lim CW (2012) Thermal effects on buckling of shear deformable nanocolumns with von Karman nonlinearity based on nonlocal stress theory. Nonlinear Anal Real World Appl 13:905–922MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lim CW (2010) On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Appl Math Mech English Edn 31:37–54MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li C, Sui SH, Chen L, Yao LQ (2018) Nonlocal elasticity approach for free longitudinal vibration of circular truncated nanocones and method of determining the range of nonlocal small scale. Smart Struct Syst 21(3):279–286Google Scholar
  25. 25.
    Kiani K (2018) Application of nonlocal higher-order beam theory to transverse wave analysis of magnetically affected forests of single-walled carbon nanotubes. Int J Mech Sci 138:1–16CrossRefGoogle Scholar
  26. 26.
    Mahmoudpour E, Hosseini-Hashemi S, Faghidian SA (2018) Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. Appl Math Model 57:302–315MathSciNetCrossRefGoogle Scholar
  27. 27.
    Li HB, Wang X, Chen JB (2018) Nonlinear dynamic responses of triple-layered graphene sheets under moving particles and an external magnetic field. Int J Mech Sci 136:413–423CrossRefGoogle Scholar
  28. 28.
    Patra AK, Gopalakrishnan S, Ganguli R (2018) Unified nonlocal rational continuum models developed from discrete atomistic equations. Int J Mech Sci 135:176–189CrossRefGoogle Scholar
  29. 29.
    Li C, Lai SK, Yang X (2019) On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter. Appl Math Model 69:127–141MathSciNetCrossRefGoogle Scholar
  30. 30.
    Li C (2014) Torsional vibration of carbon nanotubes: comparison of two nonlocal models and a semi-continuum model. Int J Mech Sci 82:25–31CrossRefGoogle Scholar
  31. 31.
    Li C, Li S, Yao LQ, Zhu ZK (2015) Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models. Appl Math Model 39:4570–4585MathSciNetCrossRefGoogle Scholar
  32. 32.
    Li C, Yao LQ, Chen WQ, Li S (2015) Comments on nonlocal effects in nano-cantilever beams. Int J Eng Sci 87:47–57CrossRefzbMATHGoogle Scholar
  33. 33.
    Shen JP, Li C (2017) A semi-continuum-based bending analysis for extreme-thin micro/nano-beams and new proposal for nonlocal differential constitution. Compos Struct 172:210–220CrossRefGoogle Scholar
  34. 34.
    Beer FP, Johnston ER Jr, Dewolf JT, Mazurek DF (2012) Mechanics of materials, 6th edn. McGraw-Hill, New YorkGoogle Scholar
  35. 35.
    Cao GX, Chen X, Kysar JW (2006) Thermal vibration and apparent thermal contraction of single-walled carbon nanotubes. J Mech Phys Solids 54:1206–1236CrossRefzbMATHGoogle Scholar
  36. 36.
    Ma HM, Gao XL, Reddy JN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56:3379–3391MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  • C. Li
    • 1
    • 2
  • N. Zhang
    • 1
  • X. L. Fan
    • 1
  • J. W. Yan
    • 3
    Email author
  • L. Q. Yao
    • 1
  1. 1.School of Rail TransportationSoochow UniversitySuzhouChina
  2. 2.State Key Laboratory of Mechanics and Control of Mechanical StructuresNanjing University of Aeronautics and AstronauticsNanjingChina
  3. 3.School of Civil Engineering and ArchitectureEast China Jiaotong UniversityNanchangChina

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