Journal of Elasticity

, Volume 130, Issue 2, pp 197–210 | Cite as

Wave Propagation in Flexoelectric Microstructured Solids

Article

Abstract

Wave propagation in elastic dielectrics with flexoelectricity, micro-inertia and strain gradient elasticity is investigated in this paper. Dispersion phenomenon, which does not exist in classical elastic dielectric theory, is observed in the flexoelectric microstructured solids. Analytical solutions for the phase velocity \(C_{p}\), group velocity \(C_{g}\) and their ratio \(\gamma = C_{g} / C_{p}\) are calculated for the case of harmonic decomposition. The magnitudes of the phase velocity and group velocity changed with the increasing of the wave number, while they are constant in the classical elastic dielectric theory. It is shown that the flexoelectricity, micro-inertia and microstructural effects are significant to predict the real behavior of longitudinal wave propagating in flexoelectric microstructured solids. Microstructural effects are not sufficient for dealing with realistic dispersion curves in flexoelectric solids, the micro-inertia and flexoelectricity are needed to obtain a physically acceptable value of the phase and group velocities.

Keywords

Wave propagation Flexoelectricity Micro-inertia Microstructure effects 

Mathematics Subject Classification

74F99 74J10 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants Nos. 11372238, 11302161, 11302162 and 11602189) and the Chang Jiang Scholar program. The support from the China Postdoctoral Science Foundation (Grant No. 2015M580835) and the Fundamental Research Funds for the Central Universities was also appreciated.

References

  1. 1.
    Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4(6), 637–642 (1968) CrossRefMATHGoogle Scholar
  2. 2.
    Suhubi, E.: Elastic dielectrics with polarization gradient. Int. J. Eng. Sci. 7(9), 993–997 (1969) CrossRefMATHGoogle Scholar
  3. 3.
    Askar, A., Lee, P., Cakmak, A.: Lattice-dynamics approach to the theory of elastic dielectrics with polarization gradient. Phys. Rev. B 1(8), 3525 (1970) ADSCrossRefGoogle Scholar
  4. 4.
    Kogan, S.M.: Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Sov. Phys., Solid State 5(10), 2069–2070 (1964) Google Scholar
  5. 5.
    Harris, P.: Mechanism for the shock polarization of dielectrics. J. Appl. Phys. 36(3), 739–741 (1965) ADSCrossRefGoogle Scholar
  6. 6.
    Tagantsev, A.: Piezoelectricity and flexoelectricity in crystalline dielectrics. Phys. Rev. B 34(8), 5883 (1986) ADSCrossRefGoogle Scholar
  7. 7.
    Zubko, P., Catalan, G., Tagantsev, A.K.: Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43, 387–421 (2013) ADSCrossRefGoogle Scholar
  8. 8.
    Cross, L.E.: Flexoelectric effects: charge separation in insulating solids subjected to elastic strain gradients. J. Mater. Sci. 41(1), 53–63 (2006) ADSCrossRefGoogle Scholar
  9. 9.
    Petrov, A.G.: Flexoelectricity of model and living membranes. Biochim. Biophys. Acta (BBA)-Biomembr. 1561(1), 1–25 (2002) CrossRefGoogle Scholar
  10. 10.
    Petrov, A.G.: Electricity and mechanics of biomembrane systems: flexoelectricity in living membranes. Anal. Chim. Acta 568(1), 70–83 (2006) CrossRefGoogle Scholar
  11. 11.
    Kalinin, S.V., Jesse, S., Liu, W., Balandin, A.A.: Evidence for possible flexoelectricity in tobacco mosaic viruses used as nanotemplates. Appl. Phys. Lett. 88(15), 153902 (2006) ADSCrossRefGoogle Scholar
  12. 12.
    Meyer, R.B.: Piezoelectric effects in liquid crystals. Phys. Rev. Lett. 22(18), 918 (1969) ADSCrossRefGoogle Scholar
  13. 13.
    Chu, B., Zhu, W., Li, N., Cross, L.E.: Flexure mode flexoelectric piezoelectric composites. J. Appl. Phys. 106(10), 4109 (2009) Google Scholar
  14. 14.
    Indenbom, V.L., Loginov, E.B., Osipov, M.A.: Flexoelectric effect and crystal structure. Kristallografiya 26(6), 1157–1162 (1981) Google Scholar
  15. 15.
    Majdoub, M., Sharma, P., Çağin, T.: Dramatic enhancement in energy harvesting for a narrow range of dimensions in piezoelectric nanostructures. Phys. Rev. B 78(12), 121407 (2008) ADSCrossRefGoogle Scholar
  16. 16.
    Lu, J., Lv, J., Liang, X., Xu, M., Shen, S.: Improved approach to measure the direct flexoelectric coefficient of bulk polyvinylidene fluoride. J. Appl. Phys. 119(9), 094104 (2016) ADSCrossRefGoogle Scholar
  17. 17.
    Zhang, S., Xu, M., Liang, X., Shen, S.: Shear flexoelectric coefficient \(\mu1211\) in polyvinylidene fluoride. J. Appl. Phys. 117(20), 204102 (2015) ADSCrossRefGoogle Scholar
  18. 18.
    Fousek, J., Cross, L., Litvin, D.: Possible piezoelectric composites based on the flexoelectric effect. Mater. Lett. 39(5), 287–291 (1999) CrossRefGoogle Scholar
  19. 19.
    Sharma, N., Maranganti, R., Sharma, P.: On the possibility of piezoelectric nanocomposites without using piezoelectric materials. J. Mech. Phys. Solids 55(11), 2328–2350 (2007) ADSCrossRefMATHGoogle Scholar
  20. 20.
    Liang, X., Hu, S., Shen, S.: Effects of surface and flexoelectricity on a piezoelectric nanobeam. Smart Mater. Struct. 23(3), 035020 (2014) ADSCrossRefGoogle Scholar
  21. 21.
    Yan, X., Huang, W.B., Kwon, S.R., Yang, S.R., Jiang, X.N., Yuan, F.G.: Design of a curvature sensor using a flexoelectric material. In: Lynch, J.P., Yun, C.B., Wang, K.W. (eds.) Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2013. Proceedings of SPIE, vol. 8692. SPIE-Int. Soc. Optical Engineering, Bellingham (2013) CrossRefGoogle Scholar
  22. 22.
    Yan, X., Huang, W., Kwon, S.R., Yang, S., Jiang, X., Yuan, F.-G.: A sensor for the direct measurement of curvature based on flexoelectricity. Smart Mater. Struct. 22(8), 085016 (2013) ADSCrossRefGoogle Scholar
  23. 23.
    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(02), 1350015 (2013) CrossRefGoogle Scholar
  24. 24.
    Maranganti, R., Sharma, N., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys. Rev. B 74(1), 014110 (2006) ADSCrossRefGoogle Scholar
  25. 25.
    Shen, S., Hu, S.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58(5), 665–677 (2010) ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Yudin, P., Tagantsev, A.: Fundamentals of flexoelectricity in solids. Nanotechnology 24(43), 432001 (2013) ADSCrossRefGoogle Scholar
  27. 27.
    Janno, J., Engelbrecht, J.: Waves in microstructured solids: inverse problems. Wave Motion 43(1), 1–11 (2005) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Georgiadis, H., Vardoulakis, I., Velgaki, E.: Dispersive Rayleigh-wave propagation in microstructured solids characterized by dipolar gradient elasticity. J. Elast. 74(1), 17–45 (2004) MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Shodja, H., Goodarzi, A., Delfani, M., Haftbaradaran, H.: Scattering of an anti-plane shear wave by an embedded cylindrical micro-/nano-fiber within couple stress theory with micro inertia. Int. J. Solids Struct. 58, 73–90 (2015) CrossRefGoogle Scholar
  30. 30.
    Papargyri-Beskou, S., Polyzos, D., Beskos, D.: Wave dispersion in gradient elastic solids and structures: a unified treatment. Int. J. Solids Struct. 46(21), 3751–3759 (2009) CrossRefMATHGoogle Scholar
  31. 31.
    Liang, X., Hu, S., Shen, S.: A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory and its applications. Compos. Struct. 111, 317–323 (2014) CrossRefGoogle Scholar
  32. 32.
    Papargyri-Beskou, S., Polyzos, D., Beskos, D.: Dynamic analysis of gradient elastic flexural beams. Struct. Eng. Mech. 15(6), 705–716 (2003) CrossRefMATHGoogle Scholar
  33. 33.
    Tsepoura, K., Papargyri-Beskou, S., Polyzos, D., Beskos, D.: Static and dynamic analysis of a gradient-elastic bar in tension. Arch. Appl. Mech. 72(6–7), 483–497 (2002) ADSCrossRefMATHGoogle Scholar
  34. 34.
    Altan, B., Evensen, H., Aifantis, E.: Longitudinal vibrations of a beam: a gradient elasticity approach. Mech. Res. Commun. 23(1), 35–40 (1996) CrossRefMATHGoogle Scholar
  35. 35.
    Chang, C.S., Gao, J.: Wave propagation in granular rod using high-gradient theory. J. Eng. Mech. 123(1), 52–59 (1997) CrossRefGoogle Scholar
  36. 36.
    Chang, C.S., Gao, J., Zhong, X.: High-gradient modeling for Love wave propagation in geological materials. J. Eng. Mech. 124(12), 1354–1359 (1998) CrossRefGoogle Scholar
  37. 37.
    Suiker, A., De Borst, R., Chang, C.: Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory. Acta Mech. 149(1–4), 161–180 (2001) CrossRefMATHGoogle Scholar
  38. 38.
    Suiker, A., Metrikine, A., De Borst, R.: Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models. Int. J. Solids Struct. 38(9), 1563–1583 (2001) CrossRefMATHGoogle Scholar
  39. 39.
    Metrikine, A.V., Askes, H.: One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 1: Generic formulation. Eur. J. Mech. A, Solids 21(4), 555–572 (2002) ADSCrossRefMATHGoogle Scholar
  40. 40.
    Rosi, G., Auffray, N.: Anisotropic and dispersive wave propagation within strain-gradient framework. Wave Motion 63, 120–134 (2016) CrossRefGoogle Scholar
  41. 41.
    Kondratev, A.I.: Precision measurements of the velocity and attenuation of ultrasound in solids. Sov. Phys. Acoust. 36(3), 470–476 (1990) Google Scholar
  42. 42.
    Savin, G.N., Lukashev, A.A., Lysko, E.M.: Elastic wave propagation in a solid with microstructure. Sov. Appl. Mech. 6(7), 725–728 (1970) ADSCrossRefGoogle Scholar
  43. 43.
    Erofeev, V.I., Rodyushkin, V.M.: Observation of the dispersion of elastic waves in a granular composite and a mathematical model for its description. Sov. Phys. Acoust. 38(6), 611–612 (1992) Google Scholar
  44. 44.
    Maranganti, R., Sharma, P.: Atomistic determination of flexoelectric properties of crystalline dielectrics. Phys. Rev. B 80(5), 054109 (2009) ADSCrossRefGoogle Scholar
  45. 45.
    Eliseev, E.A., Morozovska, A.N., Glinchuk, M.D., Blinc, R.: Spontaneous flexoelectric/flexomagnetic effect in nanoferroics. Phys. Rev. B 79(16), 165433 (2009) ADSCrossRefGoogle Scholar
  46. 46.
    Kuang, Z.: Some variational principles in electroelastic media under finite deformation. Sci. China, Ser. G, Phys. Mech. Astron. 51(9), 1390–1402 (2008) ADSCrossRefMATHGoogle Scholar
  47. 47.
    Kuang, Z.: Variational principles for generalized dynamical theory of thermopiezoelectricity. Acta Mech. 203(1–2), 1–11 (2009) CrossRefMATHGoogle Scholar
  48. 48.
    Kuang, Z.: Physical variational principle and thin plate theory in electro-magneto-elastic analysis. Int. J. Solids Struct. 48(2), 317–325 (2011) MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Toupin, R.A.: The elastic dielectric. J. Ration. Mech. Anal. 5(6), 849–915 (1956) MathSciNetMATHGoogle Scholar
  50. 50.
    McMeeking, R.M., Landis, C.M., Jimenez, S.M.A.: A principle of virtual work for combined electrostatic and mechanical loading of materials. Int. J. Non-Linear Mech. 42, 831–838 (2007) ADSCrossRefGoogle Scholar
  51. 51.
    Gao, X.-L., Park, S.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44(22), 7486–7499 (2007) CrossRefMATHGoogle Scholar
  52. 52.
    Sharma, N., Landis, C., Sharma, P.: Piezoelectric thin-film superlattices without using piezoelectric materials. J. Appl. Phys. 108(2), 024304 (2010) ADSCrossRefGoogle Scholar
  53. 53.
    Liu, C., Hu, S., Shen, S.: Effect of flexoelectricity on band structures of one-dimensional phononic crystals. J. Appl. Mech. 81(5), 051007 (2014) CrossRefGoogle Scholar
  54. 54.
    Hu, S., Shen, S.: Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China Ser. G 39(12), 1762–1769 (2010) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.State Key Laboratory for Strength and Vibration of Mechanical StructuresXi’an Jiaotong UniversityXi’anP.R. China

Personalised recommendations