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



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.


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.


Nano-beam Cantilever Nonlocal theory Dynamical load coefficient 



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).


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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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