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The Size-Dependent Electromechanical Coupling Response in Circular Micro-Plate due to Flexoelectricity

Published online by Cambridge University Press:  17 October 2016

X. Ji  and
A.-Q. Li
X. Ji
Affiliation:
School of Mechanical & Automotive Engineering, Qilu University of TechnologyJinan, China
A.-Q. Li*
Affiliation:
School of Mechanical EngineeringKey Laboratory of High Efficiency and Clean Mechanical ManufactureShandong UniversityJinan, China
*
*Corresponding author ([email protected])
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Abstract

Flexoelectricity, the coupling of strain gradient to polarization, enhances the properties of piezoelectric response desirable for advanced MEMS dramatically even in centrosymmetric dielectrics. In this paper, the general formulations of the flexoelectric couple stress theory presented by Hadjesfandiari in orthogonal curvilinear coordinate system are derived, and are then specified for the case of cylindrical coordinates. A size-dependent flexoelectric model of circular plate is established based on the current formulations in cylindrical coordinates. The governing equations, boundary conditions and initial conditions are derived by applying Hamilton's principle. The static bending and free vibration problems of a simply supported axisymmetric circular plate are carried out to illustrate the applicability of the present model. Numerical results reveal that a homogeneous electric field between the up and down surfaces of the circular plate is induced indeed. The generated deflection, induced voltage and natural frequency show obvious size effect, but the size effect is almost diminishing as the thickness of the plate is far greater than the material length scale parameter. As the increase of the flexoelectric coefficient, the induced voltage increases evidently and the generated deflection and the natural frequency increase weakly.

Keywords

Circular micro-plateFlexoelectricityCouple stressSize effect
Type
Research Article
Information
Journal of Mechanics , Volume 33 , Issue 6 , December 2017 , pp. 873 - 883
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2017 

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References

1. Zubko, P., et al., “Strain-gradient-induced polarization in SrTiO3 single crystals,” Physical Review Letters, 99, pp. 167601-1-4 (2007).Google Scholar
2. Majdoub, M. S., et al., “Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect,” Physical Review B, 77, pp. 125424-1-9 (2008).CrossRefGoogle Scholar
3. Zubko, P., et al., “Flexoelectric effect in solids,” Annual Review of Materials Research, 43, pp. 387–421 (2013).CrossRefGoogle Scholar
4. Fu, J. Y., et al., “Gradient scaling phenomenon in microsize flexoelectric piezoelectric composites,” Applied Physics Letters, 91, pp. 182910-1-3 (2007).Google Scholar
5. Lam, D. C. C., et al., “Experiments and theory in strain gradient elasticity,” Journal of the Mechanics and Physics of Solids, 51, pp. 1477–1508 (2003).CrossRefGoogle Scholar
6. Chasiotis, I. and Knauss, W. G., “The mechanical strength of polysilicon films: Part 2. Size effects associated with elliptical and circular perforations,” Journal of Mechanics and Physics of Solids, 51, pp. 1551–1572 (2003).Google Scholar
7. Tang, C. and Alici, G., “Evaluation of length-scale effects for mechanical behaviour of micro- and nanocantilevers: I. experimental determination of length-scale factors,” Journal of Physics D-Applied Physics, 44, pp. 335501-1-12 (2011).Google Scholar
8. Tang, C. and Alici, G., “Evaluation of length-scale effects for mechanical behaviour of micro- and nanocantilevers: II. experimental verification of deflection models using atomic for microscopy,” Journal of Physics D-Applied Physics, 44, pp. 335502-1-7 (2011).Google Scholar
9. Li, A. Q., et al., “A size-dependent bilayered microbeam model based on strain gradient elasticity theory,” Composite Structures, 108, pp. 259–266 (2014).Google Scholar
10. Li, A. Q., et al., “A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory,” Composite Structures, 113, pp. 272–280 (2014).Google Scholar
11. Kong, S. L., et al., “The size-dependent natural frequency of Bernoulli-Euler micro-beams,” International Journal of Engineering Science, 46, pp. 427–437 (2008).CrossRefGoogle Scholar
12. Kong, S. L., et al., “Static and dynamic analysis of micro beams based on strain gradient elasticity theory,” International Journal of Engineering Science, 47, pp. 487–498 (2009).CrossRefGoogle Scholar
13. Wang, B. L., et al., “A micro scale Timoshenko beam model based on strain gradient elasticity theory,” European Journal of Mechanics A-Solids, 29, pp. 591–599 (2010).Google Scholar
14. Wang, B. L., et al., “A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory,” European Journal of Mechanics A-Solids, 30, pp. 517–524 (2011).Google Scholar
15. Ma, W. H., “A study of flexoelectric coupling associated internal electric field and stress in thin film ferroelectrics,” Physica status Solidi (b), 245, pp. 761–768 (2008).CrossRefGoogle Scholar
16. Huang, W. B., et al., “Scaling effect of flexoelectric (Ba,Sr)TiO3 microcantilevers,” Physica Status Solidi-Rapid Research Letters, 5, pp. 350–352 (2011).Google Scholar
17. Li, Y., et al., “Enhanced flexoelectric effect in a non-ferroelectric composite,” Applied Physics Letters, 103, pp. 142909-1-4 (2013).Google Scholar
18. Baskaran, S., et al., “Experimental studies on the direct flexoelectric effect in a-phase polyvinylidene fluoride films,” Applied Physics Letters, 98, pp. 242901-1-3 (2011).Google Scholar
19. Baskaran, S., et al., “Strain gradient induced electric polarization in a-phase polyvinylidene fluoride films under bending conditions,” Journal of Applied Physics, 111, pp. 014109-1-5 (2012).Google Scholar
20. Mindlin, R. D. and Tiersten, H. F., “Effects of couple-stresses in linear elasticity,” Archive for Rational Mechanics and Analysis, 11, pp. 415–448 (1962).Google Scholar
21. Toupin, R. A., “Elastic materials with couple-stresses,” Archive for Rational Mechanics and Analysis, 11, pp. 385–414 (1962).Google Scholar
22. Koiter, W. T., “Couple stresses in the theory of elasticity, I and II,” Philosophical Transactions of the Royal Society of London B, 67, pp. 17–44 (1969).Google Scholar
23. Mindlin, R. D., “Micro-structure in linear elasticity,” Archive for Rational Mechanics and Analysis, 16, pp. 51–78 (1964).Google Scholar
24. Yang, F., et al., “Couple stress based strain gradient theory for elasticity,” International Journal of Solids And Structures, 39, pp. 2731–2743 (2002).Google Scholar
25. Hadjesfandiari, A. R. and Dargush, G. F., “Couple stress theory for solids,” International Journal of Solids And Structures, 48, pp. 2496–2510 (2011).CrossRefGoogle Scholar
26. Wang, G. F., “A piezoelectric constitutive theory with rotation gradient effects,” European Journal of Mechanics A-Solids, 23, pp. 455–466 (2004).CrossRefGoogle Scholar
27. Liang, X. and Shen, S. P., “Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity,” International Journal of Applied Mechanics, 5, pp. 1350015-1-16 (2013).Google Scholar
28. Shen, S. P. and Hu, S. L., “A theory of flexoelectricity with surface effect for elastic dielectrics,” Journal of the Mechanics and Physics of Solids, 58, pp. 665–677 (2010).Google Scholar
29. Hadjesfandiari, A. R., “Size-dependent piezoelectricity,” International Journal of Solids and Structures, 50, pp. 2781–2791 (2013).Google Scholar
30. Yan, Z. and Jiang, L. Y., “Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity,” Journal of Physics D-Applied Physics, 46, pp. 355502-1-7 (2013).Google Scholar
31. Hu, Y. T., et al., “The effects of first-order strain gradient in micro piezoelectric-bimorph power harvesters,” IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, 58, pp. 849–852 (2011).Google Scholar
32. Wang, J. N., et al., “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Materials and Structures, 21, pp. 015006-1-6 (2012).Google Scholar
33. Li, A. Q. et al., “Size-dependent analysis of a three-layer microbeam including electromechanical coupling,” Composite Structures, 116, pp. 120–127 (2014).Google Scholar
34. Zhao, J. and Pedroso, D., “Strain gradient theory in orthogonal curvilinear coordinates,” International Journal of Solids and Structures, 45, pp. 3507–3520 (2008).Google Scholar
35. Zhang, B., et al., “A novel size-dependent functionally graded curved microbeam model based on the strain gradient elasticity theory,” Composite Structures, 106, pp. 374–392 (2013).Google Scholar
36. Wang, Q., et al., “Analysis of piezoelectric coupled circular plate,” Smart Materials & Structures, 10, pp. 229–239 (2001).Google Scholar
37. Sekouri, E. M., et al., “Modeling of a circular plate with piezoelectric actuators,” Mechatronics, 14, pp. 1007–1020 (2004).Google Scholar
38. Wah, T., “Vibration of circular plates,” The Journal of the Acoustical Society of America, 34, pp. 275–281 (1962).CrossRefGoogle Scholar