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Two Differential Equations for Investigating the Vibration of Conductive Nanoplates in a Constant In-Plane Magnetic Field Based on the Energy Conservation Principle and the Local Equilibrium Equations

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Abstract

In this paper, we investigate the movement of nanoplates using two approaches: extracting its differential equation, and extracting an integral equation based on the energy conservation principle. To extract the differential equation describing the free vibration of nanoplates in constant in-plane magnetic fields, we first use the theories developed by Kirchhoff and Mendelian to investigate the deformation of the nanoplates. Then, we use Lorentz force to calculate the electromagnetic force, and we use the Eringen’s non-local theory to consider the non-local effects. The extracted equation has an exact solution for calculating the natural frequency of rectangle nanoplates with simply support boundary conditions. To extract the differential equation based on the energy conservation principle, we calculate the stresses based on local equilibrium equations. These stresses are then used to discover the relationship between inner moments and mid-plane deformation. After that, based on the energy conservation principle, an equation describing the vibration is obtained. Finally, based on the extracted equation, the curvatures are calculated so that the Eringen’s non-local theory is satisfied. These curvatures are used to calculate the elastic potential energy and rate of work done for the applied magnetic field. For a rectangular plate with simply support, the results indicate that the two equations are consistent with each other in predicting the frequency. However, as the power of the applied field increases, the existence of magnetic viscosity is predicted, and the difference between the results of these equations will become significant.

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REFERENCES

  1. I. Y. Bu, Superlatt. Microstruct. 64, 213 (2013).

    Article  CAS  Google Scholar 

  2. S. S. Bhe et al., Scr. Mater. 65, 1081 (2011).

    Article  Google Scholar 

  3. N. D. Hoa, N. van Duy, and N. van Hieu, Mater. Res. Bull. 48, 440 (2013).

    Article  CAS  Google Scholar 

  4. L. Zhong et al., Electrochim. Acta 89, 222 (2013).

    Article  CAS  Google Scholar 

  5. X. le Guével et al., J. Phys. Chem. C 113, 16380 (2009).

    Article  Google Scholar 

  6. M. Pumera, Chem. Record 9, 211 (2009).

    Article  CAS  Google Scholar 

  7. M. D. Stoller et al., Nano Lett. 8, 3498 (2008).

    Article  CAS  Google Scholar 

  8. Y. Zhong, Q. Guo, S. Li, J. Shi, and L. Liu, Sol. Energy Mater. Sol. Cells 94, 1011 (2010).

    Article  CAS  Google Scholar 

  9. J. Li et al., Nanoscale 3, 5103 (2011).

    Article  CAS  Google Scholar 

  10. D. Wang, R. Kou, D. Choi, et al., ACS Nano 4, 1587 (2010).

    Article  CAS  Google Scholar 

  11. F. Li, X. Han, and Sh. Liu, Biosens. Bioelectron. 26, 2619 (2011).

    Article  CAS  Google Scholar 

  12. L. Sun et al., J. Phys. Chem. C 112, 1415 (2008).

    Article  CAS  Google Scholar 

  13. F. Li, X. Han, and Sh. Liu, Biosens. Bioelectron. 26, 2619 (2011).

    Article  CAS  Google Scholar 

  14. H. Wang et al., Appl. Math. Model. 34, 878 (2010).

    Article  Google Scholar 

  15. X. Wang et al., Appl. Math. Model. 36, 648 (2012).

    Article  Google Scholar 

  16. S. Narendar, S. S. Gupta, and S. Gopalakrishnan, Appl. Math. Model. 36, 4529 (2012).

    Article  Google Scholar 

  17. T. Murmu, M. A. McCarthy, and S. Adhikari, J. Sound Vibrat. 331, 5069 (2012).

    Article  Google Scholar 

  18. K. Kiani, Acta Mech. 224, 3139 (2013).

    Article  Google Scholar 

  19. L. Wei, Soh Ai Kah, and R. Hu, Int. J. Mech. Sci. 49, 440 (2007).

    Article  Google Scholar 

  20. F. C. Moon and Y.-Hs. Pao, J. Appl. Mech. 35, 53 (1968).

    Article  Google Scholar 

  21. W. Yang et al., J. Appl. Mech. 66, 913 (1999).

    Article  CAS  Google Scholar 

  22. T. J. Hoffmann and M. Chudzicka-Adamczak, Int. J. Eng. Sci. 47, 735 (2009).

    Article  CAS  Google Scholar 

  23. A. C. Eringen, Int. J. Eng. Sci. 27, 363 (1989).

    Article  Google Scholar 

  24. T. Murmu, M. A. McCarthy, and S. Adhikari, Compos. Struct. 96, 57 (2013).

    Article  Google Scholar 

  25. K. Kiani, Phys. E (Amsterdam, Neth.) 57, 179 (2014).

  26. A. C. Eringen and D. G. B. Edelen, Int. J. Eng. Sci. 10, 233 (1972).

    Article  Google Scholar 

  27. A. C. Eringen, J. Appl. Phys. 54, 4703 (1983).

    Article  Google Scholar 

  28. B. Asli Beigi and P. Kameli, “The study of the magnetic property of Ferrite nanoparticles and the effect of silver and cobalt doping on their magnetic property,” Ph. D. Thesis (Isfahan Univ. of Technol., Isfahan, 2013).

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Correspondence to Saeed Shahsavari.

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Shahsavari, S., Moradi, M. & Shahidi, A. Two Differential Equations for Investigating the Vibration of Conductive Nanoplates in a Constant In-Plane Magnetic Field Based on the Energy Conservation Principle and the Local Equilibrium Equations. Nanotechnol Russia 16, 175–182 (2021). https://doi.org/10.1134/S2635167621020142

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  • DOI: https://doi.org/10.1134/S2635167621020142

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