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Analytical and Numerical Investigation of a Nonlinear Nanobeam Model

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Abstract

Purpose

This article investigates the mechanical behavior of a nonlinear nanobeam subjected to static and dynamic loading conditions using a nonlinear model that incorporates analytical and finite-element methods, and the non-local strain gradient theory.

Methods

The study derives the equation of motion for the nanobeam using Hamilton’s principle and establishes dimensionless parameters. The deflection of the nanobeam under static loading conditions is analyzed using the Galerkin method, while the time-dependent nonlinear equation is obtained using the same method under initial conditions. The natural frequency and nonlinear frequency of oscillation are determined using the method of multiple scales. The forced vibrations of the nanobeam are analyzed using the multiple scale method, and the amplitude and phase of oscillation are determined.

Results

The study examines the convergence of the finite-element method, and a comparison is made between the outcomes of two analytical and numerical methods. It investigates the role of non-local and length scale parameters on static and vibration behavior, and compares classical, non-local, and non-local strain gradient theories.

Conclusion

The results reveal that the non-local strain gradient theory demonstrates a combined softening and stiffening behavior, dependent on the dimensions of the structures, with the boundary between the two regions displayed in the results.

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Data Availability

The datasets utilized to generate the graphs and results in the current study can be acquired directly from the numerical and analytical simulation of the associated mathematical equations presented in the manuscript. The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Bhushan B (2017) Nanotribology and nanomechanics: an introduction. Springer, Berlin

    Book  Google Scholar 

  2. Li S, Wang G (2008) Introduction to micromechanics and nanomechanics. World Scientific Publishing Company, Singapore

    Book  Google Scholar 

  3. Chuang T-J, Anderson P, Wu M-K, Hsieh S (2006) Nanomechanics of materials and structures. Springer, Berlin

    Book  Google Scholar 

  4. Loh OY, Espinosa HD (2012) Nanoelectromechanical contact switches. Nat Nanotechnol 7:283–295

    Article  Google Scholar 

  5. Takahata K (2013) Advances in micro/nano electromechanical systems and fabrication technologies. BoD–Books on Demand

  6. Anđelić N, Car Z, Čanađija M (2019) NEMS resonators for detection of chemical warfare agents based on graphene sheet. Math Problems Eng 2019:1–23

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  8. Hosseini M, Hadi A, Malekshahi A, Shishesaz M (2018) A review of size-dependent elasticity for nanostructures. J Comput Appl Mech 49:197–211

    Google Scholar 

  9. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16

    Article  MathSciNet  Google Scholar 

  10. 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 

  11. Gopalakrishnan S, Narendar S (2013) Wave propagation in nanostructures: nonlocal continuum mechanics formulations. Springer, Berlin

    Book  Google Scholar 

  12. Patnaik S, Sidhardh S, Semperlotti F (2021) Towards a unified approach to nonlocal elasticity via fractional-order mechanics. Int J Mech Sci 189:105992

    Article  Google Scholar 

  13. Shaat M, Ghavanloo E, Fazelzadeh SA (2020) Review on nonlocal continuum mechanics: physics, material applicability, and mathematics. Mech Mater 150:103587

    Article  Google Scholar 

  14. Challamel N, Wang CM, Zhang H, Elishakoff I (2021) Lattice-based nonlocal elastic structural models. In: Ghavanloo E, Fazelzadeh SA, Marotti de Sciarra F (eds) Size-dependent continuum mechanics approaches: theory and applications. Springer, Cham, pp 1–50

    Google Scholar 

  15. Wang B, Deng Z, Ouyang H, Xu X (2015) Free vibration of wavy single-walled fluid-conveying carbon nanotubes in multi-physics fields. Appl Math Model 39:6780–6792

    Article  MathSciNet  Google Scholar 

  16. Hosseini M, Sadeghi-Goughari M (2016) Vibration and instability analysis of nanotubes conveying fluid subjected to a longitudinal magnetic field. Appl Math Model 40:2560–2576

    Article  MathSciNet  Google Scholar 

  17. Danesh H, Javanbakht M (2022) Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory. Math Mech Solids 27:557–577

    Article  MathSciNet  Google Scholar 

  18. Russillo AF, Failla G (2022) Wave propagation in stress-driven nonlocal Rayleigh beam lattices. Int J Mech Sci 215:106901

    Article  Google Scholar 

  19. Civalek Ö, Uzun B, Yaylı MÖ (2022) An effective analytical method for buckling solutions of a restrained FGM nonlocal beam. Comput Appl Math 41:67

    Article  MathSciNet  Google Scholar 

  20. 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  Google Scholar 

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

    Article  Google Scholar 

  22. Merzouki T, Ahmed HMS, Bessaim A, Haboussi M, Dimitri R, Tornabene F (2021) Bending analysis of functionally graded porous nanocomposite beams based on a non-local strain gradient theory. Math Mech Solids 27:66–92

    Article  MathSciNet  Google Scholar 

  23. Jin J, Hu N, Hu H (2022) Investigation of size effect on band structure of 2D nano-scale phononic crystal based on nonlocal strain gradient theory. Int J Mech Sci 219:107100

    Article  Google Scholar 

  24. Li C, Zhu CX, Zhang N, Sui SH, Zhao JB (2022) Free vibration of self-powered nanoribbons subjected to thermal-mechanical-electrical fields based on a nonlocal strain gradient theory. Appl Math Model 110:583–602

    Article  MathSciNet  Google Scholar 

  25. Boyina K, Piska R (2023) Wave propagation analysis in viscoelastic Timoshenko nanobeams under surface and magnetic field effects based on nonlocal strain gradient theory. Appl Math Comput 439:127580

    MathSciNet  Google Scholar 

  26. Faghidian SA (2021) Contribution of nonlocal integral elasticity to modified strain gradient theory. Eur Phys J Plus 136:559

    Article  Google Scholar 

  27. Apuzzo A, Barretta R, Faghidian SA, Luciano R, Marotti de Sciarra F (2018) Free vibrations of elastic beams by modified nonlocal strain gradient theory. Int J Eng Sci 133:99–108

    Article  MathSciNet  Google Scholar 

  28. Hadi A, Nejad MZ, Hosseini M (2018) Vibrations of three-dimensionally graded nanobeams. Int J Eng Sci 128:12–23

    Article  MathSciNet  Google Scholar 

  29. Sahmani S, Aghdam MM (2018) Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells. Compos B Eng 132:258–274

    Article  Google Scholar 

  30. Li L, Hu Y (2015) Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int J Eng Sci 97:84–94

    Article  MathSciNet  Google Scholar 

  31. Caporale A, Darban H, Luciano R (2022) Nonlocal strain and stress gradient elasticity of Timoshenko nano-beams with loading discontinuities. Int J Eng Sci 173:103620

    Article  MathSciNet  Google Scholar 

  32. Jiang Y, Li L, Hu Y (2023) A physically-based nonlocal strain gradient theory for crosslinked polymers. Int J Mech Sci 245:108094

    Article  Google Scholar 

  33. Babaei H (2022) Nonlinear analysis of size-dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory. Eng Comput 38:1717–1734

    Article  Google Scholar 

  34. Thai CH, Nguyen-Xuan H, Phung-Van P (2023) A size-dependent isogeometric analysis of laminated composite plates based on the nonlocal strain gradient theory. Eng Comput 39:331–345

    Article  Google Scholar 

  35. Daikh AA, Houari MSA, Belarbi MO, Mohamed SA, Eltaher MA (2022) Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory. Defence Technol 18:1778–1809

    Article  Google Scholar 

  36. Li Q, Wu D, Gao W, Hui D (2023) Nonlinear dynamic stability analysis of axial impact loaded structures via the nonlocal strain gradient theory. Appl Math Model 115:259–278

    Article  Google Scholar 

  37. Yang W, Wang S, Kang W, Yu T, Li Y (2023) A unified high-order model for size-dependent vibration of nanobeam based on nonlocal strain/stress gradient elasticity with surface effect. Int J Eng Sci 182:103785

    Article  MathSciNet  Google Scholar 

  38. Vadlamani S, Arun CO (2019) Construction of beam elements considering von Kármán nonlinear strains using B-spline wavelet on the interval. Appl Math Model 68:675–695

    Article  MathSciNet  Google Scholar 

  39. Mojahedi M, Rahaeifard M (2016) A size-dependent model for coupled 3D deformations of nonlinear microbridges. Int J Eng Sci 100:171–182

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanical and Energy Engineering, Shahid Beheshti University, Tehran, Iran

    Mahdi Mojahedi

  2. Mechanical Engineering Group, Golpayegan College of Engineering, Isfahan University of Technology, Golpayegan, Iran

    Mahdi Mojahedi

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  1. Mahdi Mojahedi
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Correspondence to Mahdi Mojahedi.

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Mojahedi, M. Analytical and Numerical Investigation of a Nonlinear Nanobeam Model. J. Vib. Eng. Technol. 12, 3471–3485 (2024). https://doi.org/10.1007/s42417-023-01058-5

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  • DOI: https://doi.org/10.1007/s42417-023-01058-5

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