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Size-dependent direct and converse flexoelectricity around a micro-hole

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Abstract

In this paper, the complete explicit solution for physical fields around a micro-hole is solved with the simultaneous consideration of the strain gradient elasticity, the direct flexoelectricity, and the converse flexoelectricity. First, the higher-order Navier-like governing equations are proposed for an isotropic flexoelectric solid by using an extended linear theory of flexoelectric materials considering the coupling between the strain gradient and the polarization, and conversely between the polarization gradient and the strain. Second, the displacement, the electric potential, and the polarization are successfully obtained by solving the corresponding boundary value problems for a micro-hole model. Finally, the influence of the strain gradients and the flexoelectric effect on the mechanical fields and the electric responses are studied around the micro-hole in flexoelectric solids, and the size effects of the direct flexoelectricity and the converse flexoelectricity are also investigated. The results indicate that an electric response can be obviously induced by a mechanical loading due to the direct flexoelectricity. Conversely, a mechanical strain can be produced by an electric field through the converse flexoelectricity. The size dependence of both the direct flexoelectricity and the converse flexoelectricity is also successfully predicted in this paper, in the form that the flexoelectricity significantly increases with the decrease in the sample size.

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References

  1. Lu, H., Bark, C.W., Esque de los Ojos, D., Alcala, J., Eom, C.B., Catalan, G., Gruverman, A.: Mechanical writing of ferroelectric polarization. Science 336, 59–61 (2012)

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Narvaez, J., Vasquez-Sancho, F., Catalan, G.: Enhanced flexoelectric-like response in oxide semiconductors. Nature 538, 219–221 (2016)

    Google Scholar 

  5. Vasquez-Sancho, F., Abdollahi, A., Damjanovic, D., Catalan, G.: Flexoelectricity in Bones. Adv. Mater. 30, 1705316 (2018)

    Google Scholar 

  6. Mashkevich, V.S., Tolpygo, K.B.: Electrical, optical and elastic properties of diamond type crystals. Sov. Phys. JETP 5, 435–439 (1957)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4, 637–642 (1968)

    MATH  Google Scholar 

  9. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)

    MATH  Google Scholar 

  10. 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)

    Google Scholar 

  11. 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)

    MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Yu, P.F., Chen, J.Y., Wang, H.L., Liang, X., Shen, S.P.: Path-independent integrals in electrochemomechanical systems with flexoelectricity. Int. J. Solids Struct. 147, 20–28 (2018)

    Google Scholar 

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

    Google Scholar 

  17. 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)

    Google Scholar 

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

    Google Scholar 

  19. Liang, X., Zhang, R., Hu, S., Shen, S.: Flexoelectric energy harvesters based on Timoshenko laminated beam theory. J. Int. Mater. Syst. Struct. 28, 2064–2073 (2017)

    Google Scholar 

  20. Mao, S., Purohit, P.K.: Defects in flexoelectric solids. J. Mech. Phys. Solids 84, 95–115 (2015)

    MathSciNet  Google Scholar 

  21. Tian, X.P., Li, Qun, Deng, Q.: The J-integral in flexoelectric solids. Int. J. Fract. 215, 67–76 (2019)

    Google Scholar 

  22. Sladek, J., Sladek, V., Wunsche, M., Zhang, C.Z.: Effects of electric field and strain gradients on cracks in piezoelectric solids. Eur. J. Mech. ASolid 71, 187–198 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Sladek, J., Sladek, V., Jus, M.: The MLPG for crack analyses in composites with flexoelectricity effects. Compos. Struct. 204, 105–113 (2018)

    Google Scholar 

  24. Abdollahi, A., Peco, C., Millán, D., Arroyo, M., Catalan, G., Arias, I.: Fracture toughening and toughness asymmetry induced by flexoelectricity. Phys. Rev. B 92, 094101 (2015)

    Google Scholar 

  25. Huang, W.B., Yan, X., Kwon, S.R., Zhang, S.J., Yuan, F.G., Jiang, X.N.: Flexoelectric strain gradient detection using Ba0.64Sr0.36TiO3 for sensing. Appl. Phys. Lett. 101, 252903 (2012)

    Google Scholar 

  26. Koester, K.J., Ager, J.W., Ritchie, R.O.: The effect of aging on crack-growth resistance and toughening mechanisms in human dentin. Biomaterials 29, 1318–1328 (2008)

    Google Scholar 

  27. Ivancik, J., Arola, D.D.: The importance of microstructural variations on the fracture toughness of human dentin. Biomaterials 34, 864–874 (2013)

    Google Scholar 

  28. Montoya, C., Arola, D., Ossa, E.A.: Importance of tubule density to the fracture toughness of dentin. Arch. Oral Biol. 67, 9–14 (2016)

    Google Scholar 

  29. 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, 7486–7499 (2007)

    MATH  Google Scholar 

  30. Aravas, N.: Plane-strain problems for a class of gradient elasticity models-a stress function approach. J. Elast. 104, 45–70 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Askar, A., Lee, P.C.Y., Cakmak, A.S.: The effect of surface curvature and discontinuity on the surface energy density and other induced fields in elastic dielectrics with polarization gradient. Int. J. Solids Struct. 7, 523–537 (1971)

    Google Scholar 

  32. Yang, X.M., Hu, Y.T., Yang, J.S.: Electric field gradient effects in anti-plane problems of polarized ceramics. Int. J. Solids Struct. 41, 6801–6811 (2004)

    MATH  Google Scholar 

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

    Google Scholar 

  34. Amanatidou, E., Aravas, N.: Mixed finite element formulations of strain-gradient elasticity Problems. Comput. Methods Appl. M. 191, 1723–1751 (2002)

    MATH  Google Scholar 

  35. Mao, S., Purohit, P.K., Aravas, N.: Mixed finite-element formulations in piezoelectricity and flexoelectricity. P. R. Soc. A Math. Phy. 472, 20150879 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Deng, F., Deng, Q., Yu, W.S., Shen, S.P.: Mixed finite elements for flexoelectric solids. J. Appl. Mech. Trans. ASME 84, 081004 (2017)

    Google Scholar 

  37. Deng, F., Deng, Q., Shen, S.P.: A three-dimensional mixed finite element for flexoelectricity. J. Appl. Mech. Trans. ASME 85, 031009 (2018)

    Google Scholar 

  38. Codony, D., Marco, O., Fernández-Méndez, S., Arias, I.: An immersed boundary hierarchical B-spline method for flexoelectricity. Comput. Methods Appl. M. 354, 750–782 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8, 231–282 (1997)

    Google Scholar 

  40. Lazar, M., Po, G.: The non-singular Green tensor of Mindlin’s anisotropic gradient elasticity with separable weak non-locality. Phys. Lett. A 379, 1538–1543 (2015)

    MathSciNet  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)

    Google Scholar 

  42. Wang, B., Gu, Y., Zhang, S., Chen, L.Q.: Flexoelectricity in solids: Progress, challenges, and perspectives. Prog. Mater. Sci. 106, 100570 (2019)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Key R&D Program of China (2017YFE0119800) and National Natural Science Foundation of China (Nos.11772245), the 111 Project (B18040), the Fundamental Research Funds for the Central Universities in China, the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JC-004), and the Slovak Science and Technology Assistance Agency registered under number SK-CN-RD-18-0005. The authors gratefully acknowledge the support of K.C. Wong Education Foundation.

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Authors and Affiliations

  1. State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Xinpeng Tian, Mengkang Xu, Qian Deng & Qun Li

  2. Department of Mechanics, Slovak Academy of Sciences, Bratislava, 984503, Slovakia

    Jan Sladek, Vladimir Sladek & Miroslav Repka

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  1. Xinpeng Tian
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  2. Mengkang Xu
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  3. Qian Deng
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  7. Qun Li
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Correspondence to Qun Li.

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Tian, X., Xu, M., Deng, Q. et al. Size-dependent direct and converse flexoelectricity around a micro-hole. Acta Mech 231, 4851–4865 (2020). https://doi.org/10.1007/s00707-020-02792-7

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