Skip to main content
Log in

Refined Analysis of Piezoelectric Microcantilevers in Gradient Electroelasticity Theory

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we present analytical solutions for the cantilever beam bending problems obtained in the non-classical electroelasticity theory with strain and electric field gradient effects. We show that considered model allows to provide the refined analysis for the electric field distribution around the supproted end of the cantilever taking into account the extended number of boundary conditions, which cannot be captured in classical models.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

REFERENCES

  1. R. Berger et al., ‘‘Thermal analysis using a micromechanical calorimeter,’’ Appl. Phys. Lett. 69, 40–42 (1996).

    Article  Google Scholar 

  2. H. Lang et al., ‘‘A chemical sensor based on a micromechanical cantilever array for the identification of gases and vapors,’’ Appl. Phys. A 66, S61–S64 (1998).

    Article  Google Scholar 

  3. Y. Zhang et al., ‘‘A review on pinciples and applications of Scanning Thermal Microscopy (SThM),’’ Adv. Funct. Mater. 54, 1900892 (2019).

  4. N. Lavrik et al., ‘‘Cantilever transducers as a platform for chemical and biological sensors,’’ Rev. Sci. Instrum. 75, 2229–2253 (2004).

    Article  Google Scholar 

  5. M. Krommer, ‘‘On the correction of the Bernoulli–Euler beam theory for smart piezoelectric beams,’’ Smart Mater. Struct. 10, 668–680 (2001).

    Article  Google Scholar 

  6. J. Yang, The Mechanics of Piezoelectreic Structures (World Scientific, London, 2006).

    Book  Google Scholar 

  7. A. Elshafei and F. Alraiess, ‘‘Modeling and analysis of smart piezoelectric beams using simple higher order shear deformation theory,’’ Smart Mater. Struct. 22, 035006 (2013).

  8. L. Landau et al., Electrodynamics of Continuous Media (Elsevier, Amsterdam, 2013).

    Google Scholar 

  9. Y. Yue et al., ‘‘Strain gradient and electric field gradient effects in piezoelectric cantilever beams,’’ J. Mech. Behavior Mater. 24, 121–127 (2015).

    Article  Google Scholar 

  10. Y. Yue et al., ‘‘Microscale size effects on the electromechanical coupling in piezoelectric material for anti-plane problem,’’ Smart Mater. Struct. 23, 125043 (2014).

  11. Y. Solyaev and S. Lurie, ‘‘Numerical predictions for the effective size-dependent properties of piezoelectric composites with spherical inclusions,’’ Compos. Struct. 202, 1099–1108 (2018).

    Article  Google Scholar 

  12. Y. Solyaev and S. Lurie, ‘‘Pure bending of a piezoelectric layer in second gradient electroelasticity theory,’’ Acta Mech. 230, 4197–4211 (2019).

    Article  MathSciNet  Google Scholar 

  13. S. Lurie and Y. Solyaev, ‘‘Anti-plane inclusion problem in the second gradient electroelasticity theory,’’ Int. J. Eng. Sci. 144, 103129 (2019).

  14. D. Iesan, ‘‘A theory of thermopiezoelectricity with strain gradient and electric field gradient effects,’’ Eur. J. Mech. A 67, 280–290 (2018).

    Article  MathSciNet  Google Scholar 

  15. S. Lurie and Y. Solyaev, ‘‘On the formulation of elastic and electroelastic gradient beam theories,’’ Continuum Mech. Thermodyn. 31, 1601–1613 (2019).

    Article  MathSciNet  Google Scholar 

  16. S. Lurie and Y. Solyaev, ‘‘Revisiting bending theories of elastic gradient beams,’’ Int. J. Eng. Sci. 126, 1–21 (2018).

    Article  MathSciNet  Google Scholar 

  17. R. Mindlin, ‘‘Micro-structure in linear elasticity,’’ Arch. Ration. Mech. Anal. 16, 51–78 (1964).

    Article  MathSciNet  Google Scholar 

  18. R. Maranganti and P. Sharma, ‘‘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,’’ J. Mech. Phys. Solids 55, 1823–1852 (2007).

    Article  Google Scholar 

  19. F. Dell’Isola et al., ‘‘Pantographic metamaterials: An example of mathematically driven design and of its technological challenges,’’ Continuum Mech. Thermodyn. 31, 851–884 (2019).

    Article  MathSciNet  Google Scholar 

  20. V. Eremeyev et al., ‘‘Flexoelectricity and apparent piezoelectricity of a pantographic micro-bar,’’ Int. J. Eng. Sci. 149, 103213 (2020).

  21. E. Lomakin et al., ‘‘Refined stress analysis in applied elasticity problems accounting for gradient effects,’’ Dokl. Phys. 64, 482–486 (2019).

    Article  Google Scholar 

Download references

Funding

This work was partially supported by the Russian Foundation for Basic Research (project nos. 18-08-00643 and 18-01-00553) and President Grant MK-2856.2019.8.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Y. Solyaev, A. Ustenko or E. Lykosova.

Additional information

(Submitted by A. V. Lapin)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Solyaev, Y., Ustenko, A. & Lykosova, E. Refined Analysis of Piezoelectric Microcantilevers in Gradient Electroelasticity Theory. Lobachevskii J Math 41, 2076–2082 (2020). https://doi.org/10.1134/S1995080220100157

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080220100157

Keywords:

Navigation