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Size-dependent electromechanical coupling in functionally graded flexoelectric nanocylinders

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Abstract

Flexoelectricity is an electromechanical coupling between polarization and strain gradient, which not only exhibits strong size dependency but is structure associated (geometry or microstructure). By the definition of flexoelectric coefficients, the flexoelectricity-related strain gradients can be generated by tailoring mechanical structures, such as the traditional designed truncated pyramid. In this work, a novel asymmetric nanocylinder is composed of functionally graded materials presented with uniform pressures on the top surface to create a relatively large inhomogeneous strain field for the achievement of obvious flexoelectric polarization. Based on the power-law-distributed material property assumption, we investigate the flexoelectricity of the proposed functionally graded nanocylinder. Based on the extended linear theory of piezoelectricity, the closed-form solutions are obtained, which can specifically characterize the size-dependent flexoelectricity. The most common setups are applied to quantify the flexoelectric response. From the numerical results, we can conclude that the electromechanical properties can be significantly influenced by the given FG configuration with graded material parameters, which can be a guideline for the design of novel flexoelectric devices.

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References

  1. 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 

  2. Sharma, N.D., Maranganti, R., Sharma, P.: On the possibility of piezoelectric nanocomposites without using piezoelectric materials. J. Mech. Phys. Solids 55, 2328–2350 (2007)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Krichen, S., Sharma, P.: Flexoelectricity: a perspective on an unusual electromechanical coupling. J. Appl. Mech. 83, 030801 (2016)

    Article  Google Scholar 

  7. Zhang, D.P., Lei, Y.J., Adhikari, S.: Flexoelectric effect on vibration responses of piezoelectric nanobeams embedded in viscoelastic medium based on nonlocal elasticity theory. Acta Mech. 229, 2379–2392 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang, W., Liang, X., Shen, S.: Electromechanical responses of piezoelectric nanoplates with flexoelectricity. Acta Mech. 226, 3097–3110 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mason, W.P.: Piezoelectricity, its history and applications. J. Acoust. Soc. Am. 70, 1561–1566 (1981)

    Article  Google Scholar 

  10. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Zubko, P., Catalan, G., Tagantsev, A.K.: Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43, 387–421 (2013)

    Article  Google Scholar 

  14. Nguyen, T.D., Mao, S., Yeh, Y.W., Purohit, P.K., McAlpine, M.C.: Nanoscale flexoelectricity. Adv. Mater. 25, 946–974 (2013)

    Article  Google Scholar 

  15. Zubko, P., Catalan, G., Buckley, A., Welche, P.R.L., Scott, J.F.: Strain-gradient-induced polarization in \(\text{ SrTiO }_{{3}}\) single crystals. Phys. Rev. Lett. 99, 167601 (2007)

    Article  Google Scholar 

  16. Nguyen, B.H., Zhuang, X., Rabczuk, T.: Numerical model for the characterization of Maxwell–Wagner relaxation in piezoelectric and flexoelectric composite material. Comput. Struct. 208, 75–91 (2018)

    Article  Google Scholar 

  17. Li. Y., Shu, L., Huang, W., Jiang, X., Wang, H.: Giantflexoelectricity in \({\rm Ba}_{0.6}{\rm Sr}_{0.4}{\rm TiO}_{{3}}/{\rm Ni}_{0.8}{\rm Zn}_{0.2}{\rm Fe}_{{2}}{\rm O}_{{4}}\) composite. Appl. Phys. Lett. 105, 162906 (2014)

  18. Huang, W., Yan, X., Kwon, S.R., Zhang, S., Yuan, F.G., Jiang, X.:Flexoelectric strain gradient detection using \({\rm Ba}_{0.64}{\rm Sr}_{0.36}{\rm TiO}_{{3}}\) for sensing. Appl. Phys. Lett. 101, 252903 (2012)

  19. Deng, Q., Liu, L., Sharma, P.: Flexoelectricity in soft materials and biological membranes. J. Mech. Phys. Solids 62, 209–227 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mohammadi, P., Liu, L.P., Sharma, P.: A theory of flexoelectric membranes and effective properties of heterogeneous membranes. J. Appl. Mech. 81, 011007 (2014)

    Article  Google Scholar 

  21. Ahmadpoor, F., Sharma, P.: Flexoelectricity in two-dimensional crystalline and biological membranes. Nanoscale 7, 16555–16570 (2015)

    Article  Google Scholar 

  22. Sharma, N.D., Landis, C.M., Sharma, P.: Piezoelectric thin-film super lattices without using piezoelectric materials. J. Appl. Phys. 108, 024304 (2010)

    Article  Google Scholar 

  23. Cross, L.E.: Flexoelectric effects: charge separation in insulating solids subjected to elastic strain gradients. J. Mater. Sci. 41, 53–63 (2006)

    Article  Google Scholar 

  24. Abdollahi, A., Peco, C., Millan, D., Arroyo, M., Arias, I.: Computational evaluation of the flexoelectric effect in dielectric solids. J. Appl. Phys. 116, 093502 (2014)

    Article  Google Scholar 

  25. Abdollahi, A., Millán, D., Peco, C., Arroyo, M., Arias, I.: Revisiting pyramid compression to quantify flexoelectricity: a three-dimensional simulation study. Phys. Rev. B 91, 104103 (2015)

    Article  Google Scholar 

  26. Biancoli, A., Fancher, C.M., Jones, J.L., Damjanovic, D.: Breaking of macroscopic centric symmetry in paraelectric phases of ferroelectric materials and implications for flexoelectricity. Nat. Mater. 14, 224 (2015)

    Article  Google Scholar 

  27. Zhou, H., Pei, Y., Hong, J., Fang, D.: Analytical method to determine flexoelectric coupling coefficient at nanoscale. Appl. Phys. Lett. 108, 101908 (2016)

    Article  Google Scholar 

  28. Deng, Q.: Size-dependent flexoelectric response of a truncated cone and the consequent ramifications for the experimental measurement of flexoelectric properties. J. Appl. Mech. 84, 101007 (2017)

    Article  Google Scholar 

  29. Yan, Z.: Exact solutions for the electromechanical responses of a dielectric nano-ring. J. Intel. Mater. Syst. Str. 28, 1140–1149 (2017)

    Article  Google Scholar 

  30. Huang, W., Kwon, S.R., Zhang, S., Yuan, F.G., Jiang, X.: A trapezoidal flexoelectric accelerometer. J. Intel. Mater. Syst. Str. 25, 271–277 (2014)

    Article  Google Scholar 

  31. Mbarki, R., Baccam, N., Dayal, K., Sharma, P.: Piezoelectricity above the Curie temperature? Combining flexoelectricity and functional grading to enable high-temperature electromechanical coupling. Appl. Phys. Lett. 104, 122904 (2014)

    Article  Google Scholar 

  32. Chu, L., Dui, G., Ju, C.: 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)

    Article  Google Scholar 

  33. Chu, L., Dui, G., Yan, Z., Zheng, Y.: Influence of flexoelectricity on electromechanical properties of functionally graded piezoelectric nanobeams based on modified couple stress theory. Int. J. Appl. Mech. 10, 1850103 (2018)

    Article  Google Scholar 

  34. Kumar, A., Kiran, R., Kumar, R., Jain, S.C., Vaish, R.: Flexoelectric effect in functionally graded materials: a numerical study. Eur. Phys. J. Plus 133, 141 (2018)

    Article  Google Scholar 

  35. Liang, X., Hu, S., Shen, S.: Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity. Smart Mater. Struct. 24, 105012 (2015)

    Article  Google Scholar 

  36. Yan, Z.: Size-dependent bending and vibration behaviors of piezoelectric circular nanoplates. Smart Mater. Struct. 25, 035017 (2016)

    Article  Google Scholar 

  37. Yan, Z., Jiang, L.Y.: Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity. J. Phys. D Appl. Phys. 46, 355502 (2013)

    Article  Google Scholar 

  38. Ma, W., Cross, L.E.: Flexoelectric polarization of barium strontium titanate in the paraelectric state. Appl. Phys. Lett. 81, 3440–3442 (2002)

    Article  Google Scholar 

  39. Ma, W., Cross, L.E.: Strain-gradient-induced electric polarization in lead zirconate titanate ceramics. Appl. Phys. Lett. 82, 3293–3295 (2003)

    Article  Google Scholar 

  40. Ma, W., Cross, L.E.: Flexoelectric effect in ceramic lead zirconate titanate. Appl. Phys. Lett. 86, 072905 (2005)

    Article  Google Scholar 

  41. Ma, W., Cross, L.E.: Flexoelectricity of barium titanate. Appl. Phys. Lett. 88, 232902 (2006)

    Article  Google Scholar 

  42. Bhaskar, U.K., Banerjee, N., Abdollahi, A., Wang, Z., Schlom, D.G., Rijnders, G., Catalan, G.: A flexoelectric microelectromechanical system on silicon. Nat. Nanotechnol. 11, 263 (2016)

    Article  Google Scholar 

  43. Koizum, M.: The concept of FGM. Ceram. Tran. 34, 3–10 (1993)

  44. Liew, K.M., Lei, Z.X., Zhang, L.W.: Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review. Compos. Struct. 120, 90–97 (2015)

    Article  Google Scholar 

  45. Wang, Y., Xu, R.Q., Ding, H.J.: Analytical solutions of functionally graded piezoelectric circular plates subjected to axisymmetric loads. Acta Mech. 215, 287–305 (2010)

    Article  MATH  Google Scholar 

  46. Xin, L., Lu, W., Yang, S., Ju, C., Dui, G.: Influence of linear work hardening on the elastic-plastic behavior of a functionally graded thick-walled tube. Acta Mech. 227, 2305–2321 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Chu, L., Dui, G.: Exact solutions for functionally graded micro-cylinders in first gradient elasticity. Int. J. Mech. Sci. 148, 366–373 (2018)

    Article  Google Scholar 

  48. Markworth, A.J., Ramesh, K.S., Parks, W.P.: Modelling studies applied to functionally graded materials. J. Mater. Sci. 30, 2183–2193 (1995)

    Article  Google Scholar 

  49. Mao, S., Purohit, P.K.: Insights into flexoelectric solids from strain-gradient elasticity. J. Appl. Mech. 81, 081004 (2014)

    Article  Google Scholar 

  50. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. An. 11, 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Courier Corporation, North Chelmsford (1964)

    MATH  Google Scholar 

  53. Yurkov, A.S., Tagantsev, A.K.: Strong surface effect on direct bulk flexoelectric response in solids. Appl. Phys. Lett. 108, 022904 (2016)

    Article  Google Scholar 

  54. 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, 094104 (2016)

    Article  Google Scholar 

  55. Giannakopoulos, A.E., Suresh, S.: Theory of indentation of piezoelectric materials. Acta Mater. 47, 2153–2164 (1999)

    Article  Google Scholar 

  56. Yan, Z., Jiang, L.Y.: Effect of flexoelectricity on the electroelastic fields of a hollow piezoelectric nanocylinder. Smart Mater. Struct. 24, 065003 (2015)

    Article  Google Scholar 

  57. Zhou, Y., Yang, X., Pan, D., Wang, B.: Improved incorporation of strain gradient elasticity in the flexoelectricity based energy harvesting from nanobeams. Physica E 98, 148–158 (2018)

    Article  Google Scholar 

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Acknowledgements

Chu L. acknowledges the support from the Fundamental Research Funds for the Central Universities (No. 2018YJS126), and Dui G. acknowledges the financial support of National Natural Science Foundation of China (No. 11772041).

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Authors and Affiliations

  1. Institute of Mechanics, Beijing Jiaotong University, Beijing, 100044, China

    Liangliang Chu & Guansuo Dui

  2. Department of Mechanical Engineering, University of Houston, Houston, TX, 77204, USA

    Liangliang Chu

  3. Applied Mechanics of Materials Laboratory, Department of Mechanical Engineering, Temple University, Philadelphia, PA, 19122, USA

    Yanbin Li

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  1. Liangliang Chu
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Correspondence to Guansuo Dui.

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Chu, L., Li, Y. & Dui, G. Size-dependent electromechanical coupling in functionally graded flexoelectric nanocylinders. Acta Mech 230, 3071–3086 (2019). https://doi.org/10.1007/s00707-019-02442-7

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  • DOI: https://doi.org/10.1007/s00707-019-02442-7

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