Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-19T00:35:40.017Z Has data issue: false hasContentIssue false
  • English
  • Français

A Nonlinear Electro-Mechanical Analysis of Nanobeams Based on the Size-Dependent Piezoelectricity Theory

Published online by Cambridge University Press:  11 July 2016

Y. T. Beni
Y. T. Beni*
Affiliation:
Faculty of EngineeringShahrekord UniversityShahrekord, Iran
*
*Corresponding author ([email protected])
Get access

Abstract

Nonlinear formulation of isotropic piezoelectric Euler-Bernoulli nano-beam is developed based on consistent size-dependent piezoelectricity theory. By considering geometrically nonlinear and axial displacement of the centroid of beam sections, basic nonlinear equations of piezoelectric nanobeam are derived using Hamilton's principle and variational method. Afterwards, in the special case for the formulation derived, hinged-hinged piezoelectric nanobeam is studied, and static deflection as well as free vibrations of the nanobeam under mechanical loads is determined. In this case, results of the linear formulation of the size-dependent theory are compared to those of the linear and nonlinear classical continuum theory.

Keywords

Nonlinear analysisPiezoelectricityFlexoelectricityConsistent couple stress theory
Type
Research Article
Information
Journal of Mechanics , Volume 33 , Issue 3 , June 2017 , pp. 289 - 301
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Huang, Y., Duan, X. F., Cui, Y. and Lieber, C. M., “Gallium nitride nanowire nanodevices,” Nano Letters, 2, pp. 101–104 (2002).Google Scholar
2. Chen, X., Xu, S.Y., Yao, N. and Shi, Y., “1.6 V nanogenerator for mechanical energy harvesting using PZT nanofibers,” Nano Letters, 10, pp. 2133–2137 (2010).Google Scholar
3. Yamano, A., Takata, K. and Kozuka, H., “Ferroelectric domain structures of 0.4-μm-thick Pb(Zr,Ti)O3 films prepared by polyvinylpyrrolidone-assisted Sol-Gel method,” Journal of Applied Physics, 111, pp. 054109-1 - 054109-5 (2102).Google Scholar
4. Xu, S. Y., Shi, Y. and Kim, S. G., “Fabrication and mechanical property of nano piezoelectric fibres,” Nanotechnology, 17, pp. 4497–4501 (2006).Google Scholar
5. Minary-Jolandan, M., Bernal, R. A., Kuljanishvili, I., Parpoil, V. and Espinosa, H. D., “Individual GaN nanowires exhibit strong piezoelectricity in 3D,” Nano Letters, 12, pp. 970–976 (2012).Google Scholar
6. Kulkarni, A. J., Zhou, M. and Ke, F. J., “Orientation and size dependence of the elastic properties of zinc oxide nanobelts,” Nanotechnology, 16, pp. 2749–2756 (2005).Google Scholar
7. Zhang, Y. H., Liu, B. and Fang, D. N., “Stress- induced phase transition and deformation behavior of BaTiO3 nanowires,” Journal of Applied Physics, 110, pp. 054109-1 - 054109-5 (2011).Google Scholar
8. Xiang, H. J., Yang, J. L., Hou, J. G. and Zhu, Q. S., “Piezoelectricity in ZnO nanowires: A first-principles study,” Applied Physics Letters, 89, pp. 223111-1 - 223111-3 (2006).Google Scholar
9. Zeighampour, H. and Tadi Beni, Y., “Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory,” Physica E, 61, pp. 28–39 (2014).Google Scholar
10. Zeighampour, H. and Tadi Beni, Y., “Analysis of conical shells in the framework of coupled stresses theory,” Internaional Journal of Engineering Scince, 81, pp. 107–122 (2014).Google Scholar
11. Dashtaki, P. M. and Tadi Beni, Y., “Effects of Casimir Force and Thermal Stresses on the Buckling of Electrostatic Nanobridges Based on Couple Stress Theory,” Arabian Journal for Science and Engineering, pp. 1–11 (2014).Google Scholar
12. Tadi Beni, Y., Mehralian, F. and Razavi, H., “Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory,” Composite Structures, 120, pp. 65–78 (2015).Google Scholar
13. Tadi Beni, Y. and Karimi Zeverdejani, M., “Free vibration of microtubules as elastic shell model based on modified couple stress theory,” Journal of Mechanics in Medicine and Biology, 15, pp. 1550037-1 - 1550037-23 (2015).Google Scholar
14. Tadi Beni, Y., Karimipour, I. and Abadyan, M., “Modeling the effect of intermolecular force on the size-dependent pull-in behavior of beam-type NEMS using modified couple stress theory,” Journal of Mechanical Science and Technology, 28, pp. 3749–3757 (2014).Google Scholar
15. Tadi Beni, Y. and Abadyan, M., “Size-dependent pull-in instability of torsional nano-actuator,” Physica Scripta, 88, pp. 055801-1 - 055801-10 (2013).Google Scholar
16. Tadi Beni, Y. and Abadyan, M., “Use of strain gradient theory for modeling the size-dependent pull-in of rotational nano-mirror in the presence of molecular force,” International Journal of Modern Physics B, 27, pp. 1350083-1 - 1350083-18 (2013).Google Scholar
17. Zeighampour, H. and Tadi Beni, Y., “Cylindrical thin-shell model based on modified strain gradient theory,” Internaional Journal of Engineering Scince, 78, pp. 27–47 (2014).Google Scholar
18. Zeverdejani, M. K. and Tadi Beni, Y., “The nano scale vibration of protein microtubules based on modified strain gradient theory,” Current Applied Physics, 13, pp. 1566–1576 (2013).Google Scholar
19. Liang, F. and Su, Y., “Stability analysis of a single-walled carbon nanotube conveying pulsating and viscous fluid with nonlocal effect,” Applied. Mathematical Modelling, 37, pp. 6821–6828 (2013).Google Scholar
20. Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R. and Sahmani, S., “Postbuckling characteristics of nanobeams based on the surface elasticity theory,” Composite Part B: Engineering, 55, pp. 240–246 (2013).CrossRefGoogle Scholar
21. Voigt, W., “Theoretische Studien fiber die Elastizitatsverhiltnisse der Kristalle (Theoretical Studies on the Elasticity Relationships of Crystals)”, Abhandlungen zur Geschichte der mathematischen Wissenschaften, 34 (1887).Google Scholar
22. Cosserat, E. and Cosserat, F., Théorie des corps déformables (Theory of Deformable Bodies), Librairie Scientifique A. Hermann et fils, Paris (1909).Google Scholar
23. Toupin, R. A., “Elastic materials with couple-stresses,” Archive for Rational Mechanics and Analysis, 11, pp. 385–414 (1962).CrossRefGoogle Scholar
24. Mindlin, R. D., Tiersten, H. F., “Effects of couple-stresses in linear elasticity,” Archive for Rational Mechanics and Analysis, 11, pp. 415–488 (1962).CrossRefGoogle Scholar
25. Koiter, W. T., “Couple stresses in the theory of elasticity, I and II,” Proceedings of the Koninklijke Nederlandse Academie van Wetenschappen. Series B: Physical Sciences, 67, pp. 17–44 (1964).Google Scholar
26. Hadjesfandiari, A. R. and Dargush, F. G., “Couple stress theory for solids,” International Journal of Solids and Structures, 48, pp. 2496–2510 (2011).Google Scholar
27. Lazar, M., Maugin, G. A. and Aifantis, E. C., “On dislocations in a special class of generalized elasticity,” Physica Status Solidi (b), 242, pp. 2365–2390 (2005).Google Scholar
28. Wang, G.-F., Yu, S.-W. and Feng, X.-Q., “A piezoelectric constitutive theory with rotation gradient effects,” European Journal of Mechanics A/Solid, 23, pp. 455–466 (2004).Google Scholar
29. Hadjesfandiari, A. R., “Size-dependent piezoelectricity,” International Journal of Solids and Structures, 50, pp. 2781–2791 (2013).Google Scholar
30. Maranganti, R., Sharma, N. D. and Sharma, P., “Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green's function solutions and embedded inclusions,” Physical Review B, 74, pp. 14110-1 - 14110-14 (2006).Google Scholar
31. Yan, Z. and Jiang, L., “Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity,” Journal of Physics D: Applied Physics, 46, pp. 355502-1 - 355502-7 (2013).Google Scholar
32. Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J. and Tong, P., “Experiments and theory in strain gradient elasticity,” Journal of the Mechanics and Physics of Solids, 51, pp. 1477–1508 (2003).Google Scholar
33. Stölken, J. and Evans, A., “A microbend test method for measuring the plasticity length scale,” Acta Materialla, 46, pp. 5109–5115 (1998).Google Scholar
34. Nix, W. D. and Gao, H., “Indentation size effects in crystalline materials: a law for strain gradient plasticity,” Journal of the Mechanics and Physics of Solids, 46, pp. 411–425 (1998).Google Scholar
35. McElhaney, K., Vlassak, J. and Nix, W., “Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments,” Journal of Materials Research, 13, pp. 1300–1306 (1998).Google Scholar
36. Chong, A. and Lam, D. C., “Strain gradient plasticity effect in indentation hardness of polymers,” Journal of Materials Research, 14, pp. 4103–4110 (1999).Google Scholar
37. Cao, Y., Nankivil, D., Allameh, S. and Soboyejo, W., “Mechanical properties of Au films on silicon substrates,” Materials and Manufacturing Processes, 22, pp. 187–194 (2007).Google Scholar
38. Son, D., Jeong, J.-H. and Kwon, D., “Film-thickness considerations in microcantilever-beam test in measuring mechanical properties of metal thin film,” Thin Solid Films, 437, pp. 182–187 (2003).Google Scholar
39. Maranganti, R. and Sharma, P., “A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (ir) relevance for nanotechnologies,” Journal of the Mechanics and Physics of Solids, 55, pp. 1823–1852 (2007).Google Scholar
40. Liu, J. F., “He's variational approach for nonlinear oscillators with high nonlinearity,” Computers and Mathematics with Applications, 58, pp. 2423–2426 (2009).Google Scholar
41. Fallah, A. and Aghdam, M. M., “Thermo- mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation,” Composites Part B, 43, pp. 1523–1530 (2012).Google Scholar
42. He, J. H., “Variational approach for nonlinear oscillators,” Chaos, Solitons Fractals, 34, pp. 1430–1439 (2007).Google Scholar
43. Xia, W., Wang, L. and Yin, L., “Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration,” International Journal of Engineering Science, 48, pp. 2044–2053 (2010).Google Scholar
44. Odibat, Z. M. and Momani, S., “Application of variational iteration method to Nonlinear differential equations of fractional order,” International Journal of Nonlinear Science and Numerical Simulations, 7, pp. 27–34 (2006).Google Scholar
45. Bildik, N. and Konuralp, A., “The use of Variational Iteration Method, Differential Transform Method and Adomian Decomposition Method for solving different types of nonlinear partial differential equations,” International Journal of Nonlinear Science and Numerical Simulations, 7, pp. 65–70 (2006).Google Scholar
46. Momani, S. and Abuasad, S., “Application of He's variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, 27, pp. 1119–1123 (2006).CrossRefGoogle Scholar
47. Catalan, G., Lubk, A., Vlooswijk, A. H. G., Snoeck, E., Magen, C., Janssens, A., Rispens, G., Rijnders, G., Blank, D. H. and Noheda, A. B., “Flexoelectric rotation of polarization in ferroelectric thin films,” Nature Materials, 10, pp. 963–967 (2011).Google Scholar
48. Bradley, T. D., Hadjesfandiari, A. R. and Dargush, G. F., “Size-dependent piezoelectricity: A 2D finite element formulation for electric field-mean curvature coupling in dielectrics,” European Journal of Mechanics A/Solid, 49, pp. 308–320 (2015).Google Scholar
49. Nowacki, W., “Mathematical models of phenomenological piezoelectricity,” Banach center publications, 15, pp. 593–607 (1985).Google Scholar
50. Tagantsev, A. K., “Piezoelectricity and flexoelectricity in crystalline dielectrics,” Physical Review B, 34, pp. 5883–5889 (1986).Google Scholar