Advertisement

Exact solutions for flexoelectric response in elastic dielectric nanobeams considering generalized constitutive gradient theories

  • Sai Sidhardh
  • M. C. RayEmail author
Article

Abstract

This paper deals with the derivation of the exact solutions for the static flexoelectric response of a simply supported dielectric nano-beam subjected to distributed mechanical and electrical loads. The governing differential equations and the boundary conditions are obtained based on the Gibbs free energy for linear dielectrics considering the strain and the electrical field gradients, and their conjugates in the form of the higher order stresses and higher order polarization fields. The trends and observations from the current study are compared with the literature. The electro-mechanical coupling observed from the current model is compared for different electrical boundary conditions. The polarization and the electric field profiles across the thickness, developed due to the direct effect are also presented. Due to the use of gradient field energies, and a subsequent evaluation of their conjugates, the size effects are better exhibited by the current model than the models in the literature derived without considering strain and electric field gradients. The present study suggests that upon considering strain gradient elasticity the sensitive nature of flexoelectric nanosensors, nano energy harvesters and nanoactuators is realized. The exact solutions developed in this paper may be used as benchmark solutions for further research on flexoelectric solids.

Keywords

Strain gradient elasticity Nanobeam Flexoelectric solids Exact solutions Nanosensors 

References

  1. Abdollahi, A., Peco, C., Millán, D., Arroyo, M., Arias, I.: Computational evaluation of the flexoelectric effect in dielectric solids. J. Appl. Phys. 116(9), 093502 (2014)CrossRefGoogle Scholar
  2. Choi, S.-B., Kim, G.-W.: Measurement of flexoelectric response in polyvinylidene fluoride films for piezoelectric vibration energy harvesters. J. Phys. D Appl. Phys. 50(7), 075502 (2017)CrossRefGoogle Scholar
  3. Cross, L.E.: Flexoelectric effects: charge separation in insulating solids subjected to elastic strain gradients. J. Mater. Sci. 41(1), 53–63 (2006)CrossRefGoogle Scholar
  4. Fleck, N.A., Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41(12), 1825–1857 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 296–361 (1997)zbMATHGoogle Scholar
  6. Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49(10), 2245–2271 (2001)CrossRefzbMATHGoogle Scholar
  7. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1994)CrossRefGoogle Scholar
  8. Gao, X.-L., Park, S.K.: 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)CrossRefzbMATHGoogle Scholar
  9. Giannakopoulos, A.E., Suresh, S.: Theory of indentation of piezoelectric materials. Acta Mater. 47(7), 2153–2164 (1999)CrossRefGoogle Scholar
  10. Harris, P.: Mechanism for the shock polarization of dielectrics. J. Appl. Phys. 36(3), 739–741 (1965)CrossRefGoogle Scholar
  11. Hu, S., Shen, S.: Electric field gradient theory with surface effect for nano-dielectrics. CMC Comput. Mater. Continua 13(1), 63–88 (2009)Google Scholar
  12. Hu, S., Shen, S.: Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China Phys. Mech. Astron. 53(8), 1497–1504 (2010)CrossRefGoogle Scholar
  13. Iesan, D.: A theory of thermopiezoelectricity with strain gradient and electric field gradient effects. Eur. J. Mech. Solids 67, 280–290 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kogan, S.M.: Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Sov. Phys. Solid State 5, 2069–2070 (1964)Google Scholar
  15. Liang, X., Hu, S., Shen, S.: Effects of surface and flexoelectricity on a piezoelectric nanobeam. Smart Mater. Struct. 23(3), 035020 (2014)CrossRefGoogle Scholar
  16. Liang, X., Zhang, R., Hu, S., Shen, S.: Flexoelectric energy harvesters based on Timoshenko laminated beam theory. J. Intell. Mater. Syst. Struct. 1045389X16685438 (2017)Google Scholar
  17. Ma, W., Cross, L.E.: Observation of the flexoelectric effect in relaxor pb (mg 1/3 nb 2/3) o 3 ceramics. Appl. Phys. Lett. 78(19), 2920–2921 (2001)CrossRefGoogle Scholar
  18. Mao, S., Purohit, P.K., Aravas, N: Mixed finite-element formulations in piezoelectricity and flexoelectricity. In: Proc. R. Soc. A, vol. 472. The Royal Society (2016)Google Scholar
  19. Maranganti, R., Sharma, P.: Length scales at which classical elasticity breaks down for various materials. Phys. Rev. Lett. 98(19), 195504 (2007)CrossRefGoogle Scholar
  20. Maranganti, R., Sharma, N.D., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Greens function solutions and embedded inclusions. Phys. Rev. B 74(1), 014110 (2006)CrossRefGoogle Scholar
  21. Mashkevich, V.S.: Electrical, optical, and elastic properties of diamond-type crystals ii. lattice vibrations with calculation of atomic dipole moments. Sov. Phys. JETP 5(4) (1957)Google Scholar
  22. Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4(6), 637–642 (1968)CrossRefzbMATHGoogle Scholar
  23. Qi, L., Zhou, S., Li, A.: Size-dependent bending of an electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect. Compos. Struct. 135, 167–175 (2016)CrossRefGoogle Scholar
  24. Ray, M.C.: Exact solutions for flexoelectric response in nanostructures. J. Appl. Mech. 81(9), 091002 (2014)CrossRefGoogle Scholar
  25. Ray, M.C.: Analysis of smart nanobeams integrated with a flexoelectric nano actuator layer. Smart Mater. Struct. 25(5), 055011 (2016)CrossRefGoogle Scholar
  26. Ray, M.C.: Mesh free model of nanobeam integrated with a flexoelectric actuator layer. Compos. Struct. 159, 63–71 (2017)CrossRefGoogle Scholar
  27. Sharma, N.D., Maranganti, R., Sharma, P.: On the possibility of piezoelectric nanocomposites without using piezoelectric materials. J. Mech. Phys. Solids 55(11), 2328–2350 (2007)CrossRefzbMATHGoogle Scholar
  28. Sharma, N.D., Landis, C.M., Sharma, P.: Piezoelectric thin-film superlattices without using piezoelectric materials. J. Appl. Phys. 108(2), 024304 (2010)CrossRefGoogle Scholar
  29. Shen, S., Hu, S.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58(5), 665–677 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Sidhardh, S., Ray, M.C.: Exact solutions for elastic response in micro and nano-beams considering strain gradient elasticity. Math. Mech. Solids (2018).  https://doi.org/10.1177/1081286518761182 zbMATHGoogle Scholar
  31. Toupin, R.A.: The elastic dielectric. J. Ration. Mech. Anal. 5(6), 849–915 (1956)MathSciNetzbMATHGoogle Scholar
  32. Yan, Z.: Exact solutions for the electromechanical responses of a dielectric nano-ring. J. Intell. Mater. Syst. Struct. 28(9), 1140–1149 (2017)CrossRefGoogle Scholar
  33. Yan, Z., Jiang, L.: Surface effects on the electromechanical coupling and bending behaviours of piezoelectric nanowires. J. Phys. D Appl. Phys. 44(7), 075404 (2011)CrossRefGoogle Scholar
  34. Yan, Z., Jiang, L.Y.: Flexoelectric effect on the electroelastic responses of bending piezoelectric nanobeams. J. Appl. Phys. 113(19), 194102 (2013a)CrossRefGoogle Scholar
  35. Yan, Z., Jiang, L.: Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity. J. Phys. D Appl. Phys. 46(35), 355502 (2013b)CrossRefGoogle Scholar
  36. Yan, Z., Jiang, L.: Effect of flexoelectricity on the electroelastic fields of a hollow piezoelectric nanocylinder. Smart Mater. Struct. 24(6), 065003 (2015)CrossRefGoogle Scholar
  37. Yang, W., Liang, X., Shen, S.: Electromechanical responses of piezoelectric nanoplates with flexoelectricity. Acta Mech. 226(9), 3097–3110 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. Yudin, P.V., Tagantsev, A.K.: Fundamentals of flexoelectricity in solids. Nanotechnology 24(43), 432001 (2013)CrossRefGoogle Scholar
  39. Yurkov, A.S.: Elastic boundary conditions in the presence of the flexoelectric effect. JETP Lett. 94(6), 455–458 (2011)CrossRefGoogle Scholar
  40. Zhang, Z., Yan, Z., Jiang, L.: Flexoelectric effect on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate. J. Appl. Phys. 116(1), 014307 (2014)CrossRefGoogle Scholar
  41. Zhang, R., Liang, X., Shen, S.: A timoshenko dielectric beam model with flexoelectric effect. Meccanica 51(5), 1181–1188 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zhou, S., Li, A., Wang, B.: A reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials. Int. J. Solids Struct. 80, 28–37 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of TechnologyKharagpurIndia

Personalised recommendations