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Analysis of a curved Timoshenko nano-beam with flexoelectricity

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Abstract

A Timoshenko beam model is applied for the analysis of the flexoelectric effect for a curved nano-beam. This theory transforms the beam to a 1-d problem along the beam axis. The electric intensity vector for the open-circuit condition can be expressed by mechanical quantities, namely strains and strain gradients. The variational principle is applied to derive the system of ordinary differential equations (ODEs) for the beam deflection, cross-section rotation, and in-plane displacement. A numerical solution based on the Taylor series expansion with Cartesian coordinate along the beam is proposed for the system of ODEs. A convergence analysis with respect to the number of terms in the Taylor series expansion is performed. Numerical results for the beam deflection, rotation, in-plane displacement, and induced electric intensity vector are presented for various flexoelectric coefficients and the beam curvature.

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References

  1. Deng, Q., Kammoun, M., Erturk, A., Sharma, P.: Nanoscale flexoelectric energy harvesting. Int. J. Solids Struct. 51, 3218–3225 (2014)

    Article  Google Scholar 

  2. Faroughi, S., Rojas, E.F., Abdelkefi, A., Park, Y.H.: Reduced-order modeling and usefulness of non-uniform beams for flexoelectric energy harvesting applications. Acta Mech. 230, 2339–2361 (2019)

    Article  MathSciNet  Google Scholar 

  3. Tagantsev, A.K.: Piezoelectricity and flexoelectricity in crystalline dielectrics. Phys. Rev. B 34, 5883–5889 (1986)

    Article  Google Scholar 

  4. Tagantsev, A.K., Meunier, V., Sharma, P.: Novel electromechanical phenomena at the nanoscale: phenomenological theory and atomistic modelling. MRS Bull. 34, 643–647 (2009)

    Article  Google Scholar 

  5. Maranganti, R., Sharma, N.D., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys. Rev. B 74, 014110 (2006)

    Article  Google Scholar 

  6. Tagantsev, A.: Theory of flexoelectric effect in crystals. JETP Lett. 88, 2108–2122 (1985)

    Google Scholar 

  7. Yudin, P., Tagantsev, A.: Fundamentals of flexoelectricity in solids. Nanotechnology 24, 432001 (2013)

    Article  Google Scholar 

  8. Jiang, X., Huang, W., Zhang, S.: Flexoelectric nano-generator: materials, structures and devices. Nano Energy 2, 1079–1092 (2013)

    Article  Google Scholar 

  9. Hu, S., Shen, S.: Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China Phys. Mech. Astron. 53, 1497–1504 (2010)

    Article  Google Scholar 

  10. Wang, K.F., Wang, B.L.: An analytical model for nanoscale unimorph piezoelectric energy harvesters with flexoelectric effect. Compos. Struct. 153, 253–261 (2016)

    Article  Google Scholar 

  11. Majdoub, M.S., Sharma, P., Cagin, T.: Enhanced size dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys. Re. B 77, 125424 (2008)

    Article  Google Scholar 

  12. Liang, X., Shen, S.: Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity. Int. J. Appl. Mech. 5, 1350015 (2013)

    Article  Google Scholar 

  13. Zhang, R., Shen, S.: A Timoshenko dielectric beam model with flexoelectric effect. Meccanica 51, 1181–1188 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deng, Q., Shen, S.: The flexodynamic effect on nanoscale flexoelectric energy harvesting: a computational approach. Smart Material Struct. 27, 105001 (2018)

    Article  Google Scholar 

  15. Ma, H., Gao, X.L., Reddy, J.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Timoshenko, S.P.: Strength of materials Part I: Elementary theory and problems, 3rd edn. Van Nostrand, Princeton (1955)

    MATH  Google Scholar 

  17. Markus, S., Nanasi, T.: Vibrations of curved beams. Shock Vib. Dig. 13, 3–14 (1981)

    Article  Google Scholar 

  18. Childamparam, P., Leissa, A.W.: Vibrations of planar curved beams, rings and arches. Appl. Mech. Rev. ASME 46, 467–483 (1993)

    Article  Google Scholar 

  19. Auciello, N.M., De Rosa, M.A.: Free vibrations of circular arches: a review. J. Sound Vib. 176, 433–458 (1994)

    Article  MATH  Google Scholar 

  20. Barretta, R., de Sciarra, F.M., Vaccaro, M.S.: On nonlocal mechanics of curved elastic beams. Int. J. Eng. Sci. 44, 103140 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Alfosail, F.K., Hajjaj, A.Z., Younis, M.I.: Theoretical and experimental investigation of two-to-one internal resonance in MEMS arch resonators. J. Comput. Nonlinear Dyn. Trans. 14, 011001 (2019)

    Article  Google Scholar 

  22. Dantas, W.G., Gusso, A.: Analysis of the chaotic dynamics of MEMS/NEMS doubly clamped beam resonators with two-sided electrodes. Int. J. Bifurc. Chaos 28, 1850122 (2018)

    Article  MathSciNet  Google Scholar 

  23. Frangi, A., Gobat, G.: Reduced order modelling of the non-linear stiffness in MEMS resonators. Int. J. Non-Linear Mech. 116, 211–218 (2019)

    Article  Google Scholar 

  24. Nikpourian, A., Ghazavi, M.R., Azizi, S.: Size-dependent secondary resonance of a piezoelectrically laminated bistable MEMS arch resonator. Compos. B 173, 106850 (2019)

    Article  Google Scholar 

  25. Ouakad, H.M., Sedighi, H.M.: Static response and free vibration of MEMS arches assuming out-of-plane actuation pattern. Int. J. Non-Linear Mech. 110, 44–57 (2019)

    Article  Google Scholar 

  26. Sieberer, S., McWilliam, S., Popov, A.A.: Nonlinear electrostatic effects in MEMS ring-based rate sensors under shock excitation. Int. J. Mech. Sci. 157–158, 485–497 (2019)

    Article  Google Scholar 

  27. Wang, Z., Ren, J.: Three-to-one internal resonance in MEMS arch resonators. Sensors 19, 1888 (2019)

    Article  Google Scholar 

  28. Hosseini, S.A.H., Rahmani, O.: Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model. Appl. Phys. A 122, 169 (2016)

    Article  Google Scholar 

  29. Ebrahimi, F., Barati, M.R.: Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory. Mech. Adv. Mater. Struct. 25, 350–359 (2018)

    Article  Google Scholar 

  30. Karami, B., Shahsavari, D., Janghorban, M., Li, L.: Influence of homogenization schemes on vibration of functionally graded curved microbeams. Compos. Struct. 216, 67–79 (2019)

    Article  Google Scholar 

  31. She, G.L., Ren, Y.R., Yan, K.M.: On snap-buckling of porous FG curved nanobeams. Acta Astronaut. 161, 475–484 (2019)

    Article  Google Scholar 

  32. She, G.L., Yuan, F.G., Karami, B., Ren, Y.R., Xiao, W.S.: On nonlinear bending behavior of FG porous curved nanotubes. Int. J. Eng. Sci. 135, 58–74 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sobhy, M., Abazid, M.A.: Dynamic and instability analyses of FG graphene-reinforced sandwich deep curved nanobeams with viscoelastic core under magnetic field effect. Compos. B 174, 106966 (2019)

    Article  Google Scholar 

  34. Arefi, M., Rabczuk, T.: A nonlocal higher order shear deformation theory for electro-elastic analysis of a piezoelectric doubly curved nano shell. Compos. B 168, 496–510 (2019)

    Article  Google Scholar 

  35. Barretta, R., Caporale, A., Fagihidian, S.A., Luciano, R., de Sciarra, F.M., Medaglia, C.M.: A stress-driven local-nonlocal mixture model for Timoshenko nano-beams. Compos. B 164, 590–598 (2019)

    Article  Google Scholar 

  36. Qatu, M.S.: Theories and analyses of thin and moderately thick laminated composite curved beams. Int. J. Solids Struct. 30, 2743–2756 (1993)

    Article  MATH  Google Scholar 

  37. Lim, C.W., Wang, C.M., Kitipornchai, S.: Timoshenko curved beam bending of Euler-Bernoulli solutions. Arch. Appl. Mech. 67, 179–190 (1997)

    Article  MATH  Google Scholar 

  38. Hu, S.L., Shen, S.P.: Electric field gradient theory with surface effect for nano-dielectrics. CMC Comput. Mater. Contin. 13, 63–87 (2009)

    Google Scholar 

  39. Shen, S.P., Hu, S.L.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58, 665–677 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gitman, I., Askes, H., Kuhl, E., Aifantis, E.: Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. Int. J. Solids Struct. 47, 1099–1107 (2010)

    Article  MATH  Google Scholar 

  41. Yaghoubi, S.T., Mousavi, S.M., Paavola, J.: Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity. Int. J. Solids Struct. 109, 84–92 (2017)

    Article  Google Scholar 

  42. Deng, F., Deng, Q., Yu, W., Shen, S.: Mixed finite elements for flexoelectric solids. J. Appl. Mech. 84, 0810041-12 (2017)

    Article  Google Scholar 

  43. Sladek, J., Sladek, V., Wunsche, M., Zhang, Ch.: Effects of electric field and strain gradients on cracks in piezoelectric solids. Eur. J. Mech. A Solids 71, 187–198 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tian, X., Sladek, J., Sladek, V., Deng, Q., Li, Q.: Collocation mixed finite elements for flexoelectric solids. Int. J. Solids Struct. (submitted) (2020)

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Acknowledgements

The authors acknowledge the supports by the Slovak Science and Technology Assistance Agency registered under number SK-CN-RD-18-0005, VEGA-2/0061/20.

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Authors and Affiliations

  1. Institute of Construction and Architecture, Slovak Academy of Sciences, 84503, Bratislava, Slovakia

    Jan Sladek & Vladimir Sladek

  2. Industrial Engineering Department, Ferdowsi University of Mashhad, PO Box 91775-1111, Mashhad, Iran

    Seyed Mahmoud Hosseini

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  1. Jan Sladek
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  2. Vladimir Sladek
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  3. Seyed Mahmoud Hosseini
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Correspondence to Jan Sladek.

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Sladek, J., Sladek, V. & Hosseini, S.M. Analysis of a curved Timoshenko nano-beam with flexoelectricity. Acta Mech 232, 1563–1581 (2021). https://doi.org/10.1007/s00707-020-02901-6

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  • DOI: https://doi.org/10.1007/s00707-020-02901-6

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