Skip to main content
SpringerLink
Log in

Vibration of Two-Dimensional Functionally Graded Beam with Dynamic Flexoelectric Effect

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

In this manuscript, a functionally graded beam model considering both flexoelectric and dynamic flexoelectric effects was established. Based on the modified strain gradient theory, the governing equation was reformulated, which included the effects of the dynamic flexoelectric coefficient and the material length scale parameters. Using the electrical open circuit condition, a analytical solutions of the lateral displacement and axial displacement under static bending were obtained. In order to further analyze the influence of dynamic flexoelectric effect on the natural frequency of free vibration, the frequency equation was obtained from Navier method. Numerical results show that the electric potential, polarization, and energy efficiency of functionally graded beam can be controlled by adjusting the gradient index and material length scale parameters. In addition, the simulation results also show that the dynamic flexoelectric effect coefficient has a more significant effect on the natural frequency at small span ratios. This work can provide helpful guidance for designing energy harvesting devices in functionally graded materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig.4
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.

Similar content being viewed by others

REFERENCE

  1. P. V. Yudin and A.K. Tagantsev, “Fundamentals of flexoelectricity in solids,” Nanotechn. 43, 432001 (2013). https://doi.org/10.1088/0957-4484/24/43/432001

  2. H. G. Craighead, “Nanoelectromechanical systems,” Science 5496, 1532-5 (2000). https://doi.org/10.1126/science.290.5496.1532

    Article  ADS  Google Scholar 

  3. J. Shen, H. Wang, and S. Zheng, “Size-dependent pull-in analysis of a composite laminatedmicro-beam actuated by electrostatic and piezoelectric forces: Generalized differential quadrature method,” Int. J. Mech. Sci. 353–361 (2018). https://doi.org/10.1016/j.ijmecsci.2017.11.002

  4. A. K. Tagantsev, “Piezoelectricity and flexoelectricity in crystalline dielectrics,” Phys. Rev. B. 8, 5883–5889 (1986). https://doi.org/10.1103/PhysRevB.34.5883

    Article  ADS  Google Scholar 

  5. L. E. Cross, “Flexoelectric effects: Charge separation in insulating solids subjected to elastic strain gradients,” J. Mater Sci. 1, 53–63 (2006). https://doi.org/10.1007/s10853-005-5916-6

    Article  ADS  Google Scholar 

  6. S. Krichen and P. Sharma, “Flexoelectricity: a perspective on an unusual electromechanical coupling,” J. Appl. Mech. -T ASME 3, 030801 (2016). https://doi.org/10.1115/1.4032378

  7. A. Askar, P.C.Y. Lee, and A. S. Cakmak, “Lattice-dynamics approach to the theory of elastic dielectrics with polarization gradient,” Phys. Rev. B. 8, 3525–37 (1970). https://doi.org/10.1103/PhysRevB.1.3525

    Article  ADS  Google Scholar 

  8. C. Shen, Y. Kong, T.J. Lu, et al., “Localization of elastic waves in one-dimensional detuned phononic crystals with flexoelectric effect,” Int. J. Smart Nano Mat. 1–19 (2022). https://doi.org/10.1080/19475411.2022.2069875

  9. L.-L. Ke, J. Yang, S. Kitipornchai, et al., “Axisymmetric postbuckling analysis of size-dependent functionally graded annular microplates using the physical neutral plane,” Int J. Eng Sci. 81, 66–81 (2014). https://doi.org/10.1016/j.ijengsci.2014.04.005

    Article  MathSciNet  MATH  Google Scholar 

  10. L. S. Ma and D. W. Lee, “Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading,” Eur J. Mech A-Solids. 1, 13–20 (2012). https://doi.org/10.1016/j.euromechsol.2011.06.016

    Article  MathSciNet  MATH  Google Scholar 

  11. M. J. Khoshgoftar, A. G. Arani, and M. Arefi, “Thermoelastic analysis of a thickwalled cylinder made of functionally graded piezoelectric material,” Smart Mater Struct. 11, (2009). https://doi.org/10.1088/0964-1726/18/11/115007

  12. M. Koizumi, “The concept of FGM,” Ceramic Trans. Funct. Gradient Mater. 34, 3–10 (1993).

    Google Scholar 

  13. V. Birman and L.W. Byrd, “Modeling and analysis of functionally graded materials and structures,” Appl. Mech. Rev. 60 (1–6), 195–216 (2007). https://doi.org/10.1115/1.2777164

  14. M. Arefi, M. Pourjamshidian, A. G. Arani, et al., “Influence of flexoelectric, small-scale, surface and residual stress on the nonlinear vibration of sigmoid, exponential and power- law FG Timoshenko nano-beams,” J. Low Freq. Noise A1, 122–142 (2019). https://doi.org/10.1177/1461348418815410

    Article  Google Scholar 

  15. Y. Chen, M. Zhang, Y. Su, et al., “Coupling analysis of flexoelectric effect on functionally graded piezoelectric cantilever nanobeams,” Micromachines-Basel 6, 595–607 (2021). https://doi.org/10.3390/mi12060595

    Article  Google Scholar 

  16. S. Xiang, K. Y. Lee, and X. F. Li, “Elasticity solution of functionally graded beams with consideration of the flexoelectric effect,” J. Phys. D Appl. Phys. 10, 105301 (2020). https://doi.org/10.1088/1361-6463/ab5cc1

  17. L. Qi, “Energy harvesting properties of the functionally graded flexoelectric microbeam energy harvesters,” Energy 171, 721–730 (2019). https://doi.org/10.1016/j.energy.2019.01.047

    Article  Google Scholar 

  18. Y. H. Chen, M. M. Zhang, Z. D. Zhou, et al., “Flexoelectric effect on bending response of functionally graded piezoelectric sensors,” in Proc. of the 15th Symposium on Piezoelectrcity, Acoustic Waves and Device Applications (SPAWDA), Zhengzhou, China, Apr 16–19, 2021 (Henan Poly. Univ., Zhengzhou, 2021), pp. 20–21.

  19. R. D. Mindlin, “Micro-structure in linear elasticity,” Arch. Ration Mech. An. 1, 51–57 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  20. R. D. Mindlin and N.N. Eshel, “On first strain-gradient theories in linear elasticity,” Int. J. Solids Struct. 1, 109–124 (1968).

    Article  MATH  Google Scholar 

  21. D. C. C. Lam, F. Yang, A. C. M. Chong, et al., “Experiments and theory in strain gradient elasticity,” J. Mech. Phys. Solids 8, 1477–1508 (2003). https://doi.org/10.1016/s0022-5096(03)00053-x

    Article  ADS  MATH  Google Scholar 

  22. F. Yang, A. C. M. Chong, D. C. C. Lam, et al., “Couple stress based strain gradient theory for elasticity,” Int. J. Solids Struct. 10, 2731–2743 (2002). https://doi.org/10.1016/s0020-7683(02)00152-x

    Article  MATH  Google Scholar 

  23. S. Zhou, A. Li, and B. Wang, “A reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials,” Int. J. Solids Struct. 80, 28–37 (2016). https://doi.org/10.1016/j.ijsolstr.2015.10.018

    Article  Google Scholar 

  24. A. Li, S. Zhou, L. Qi, et al., “A reformulated flexoelectric theory for isotropic dielectrics,” J. Phys. D Appl. Phys. 46, 465502 (2015). https://doi.org/10.1088/0022-3727/48/46/465502

  25. X. Zhao, S. Zheng, Z. Li, “Bending, free vibration and buckling analyses of AFG flexoelectric nanobeams based on the strain gradient theory,” Mech. Adv. Mater. Struc. 4, 548–563 (2022). https://doi.org/10.1080/15376494.2020.1779880

    Article  Google Scholar 

  26. L. Chu, G. Dui, C. Ju, “Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory,” Compos. Struct. 186, 39–49 (2018). https://doi.org/10.1016/j.compstruct.2017.10.083

    Article  Google Scholar 

  27. Q. Chen, S. Zheng, Z. Li, et al., “Size-dependent free vibration analysis of functionally graded porous piezoelectric sandwich nanobeam reinforced with graphene platelets with consideration of flexoelectric effect,” Smart Mater. Struct. 3, 035008 (2021). https://doi.org/10.1088/1361-665X/abd963

  28. S. Hu and S. Shen, “Variational principles and governing equations in nano-dielectrics with the flexoelectric effect,” Sci. China Phys. Mech. 8, 1497–504 (2010). https://doi.org/10.1007/s11433-010-4039-5

    Article  Google Scholar 

  29. R. D. Mindlin, “Polarization gradient in elastic dielectrics,” Int J. Solids Struct. 6, 637–642 (1968). https://doi.org/10.1016/0020-7683(68)90079-6

    Article  MATH  Google Scholar 

  30. A. Kvasov and A. K. Tagantsev, “Dynamic flexoelectric effect in perovskites from first-principles calculations,” Phys. Rev. B. 5, 054104 (2015). https://doi.org/10.1103/PhysRevB.92.054104

  31. P. V. Yudin and A. K. Tagantsev, “Fundamentals of flexoelectricity in solids,” Nanotechnology 24 (43), 432001 (2013). https://doi.org/10.1088/0957-4484/24/43/432001

  32. Z. Yan and L. Jiang, “Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity,” J. Phys. D Appl. Phys. 35, 355502 (2013). https://doi.org/10.1088/0022-3727/46/35/355502

Download references

Funding

This work was supported by the National Natural Science Foundation of China (grant no. 11902076 and 11902047), the Natural Science Foundation of Fujian Province (grant no. 2019J01634).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Fuzhou University, 350002, Fuzhou, Fujian, China

    Haowei Zhang, Weifeng Leng, Yaohong Suo & Pengfei Yu

  2. School of Civil Engineering, Chang’an University, 710061, Xi’an, China

    Hailong Wang

Authors
  1. Haowei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weifeng Leng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hailong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yaohong Suo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Pengfei Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pengfei Yu.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Leng, W., Wang, H. et al. Vibration of Two-Dimensional Functionally Graded Beam with Dynamic Flexoelectric Effect. Mech. Solids 57, 1534–1549 (2022). https://doi.org/10.3103/S0025654422060140

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654422060140

Keywords:

Search

Navigation