Effect of flexoelectricity on the electromechanical response of graphene nanocomposite beam

  • S. I. KundalwalEmail author
  • K. B. Shingare
  • Ankit Rathi


Owing to its unique multifunctional and scale-dependent physical properties, graphene is emerged as promising reinforcement to enhance the overall response of nanotailored composite materials. Most recently, the piezoelectricity phenomena in graphene sheets was found through interplay between different non-centrosymmetric pores, curvature and flexoelectricity phenomena. This has added new multifunctionality to existing graphene and it seems the use of piezoelectric graphene in composites has yet to be fully explored. In this article, the mechanics of materials and finite element models were developed to predict the effective piezoelectric and elastic (piezoelastic) properties of the graphene reinforced nanocomposite material (GRNC). An analytical model based on the linear piezoelectricity and Euler beam theories was also developed to investigate the electromechanical response of GRNC cantilever beam under both electrical and mechanical loads accounting the flexoelectric effect. Furthermore, molecular dynamics simulations were carried out to determine the elastic properties of graphene which were used to develop the analytical and numerical models herein. The current results reveal that the flexoelectric effect on the elastic behavior of bending of nanocomposite beams is significant. The electromechanical behavior of GRNC cantilever beam can be tailored to achieve the desired response via a number of ways such as by varying the volume fraction of graphene layer and the application of electrical load. Our fundamental study highlights the possibility of developing lightweight and high performance piezoelectric graphene based nanoelectromechanical systems such as sensors, actuators, switches and smart electronics as compared with the existing heavy, brittle and toxic piezoelectric materials.


Graphene Flexoelectricity Piezoelectricity Nanocomposite Micromechanics Finite element 



The authors gratefully acknowledge the financial support provided by the Indian Institute of Technology Indore and the Science Engineering Research Board (SERB), Department of Science and Technology, Government of India. S.I.K. acknowledges the generous support of the SERB Early Career Research Award Grant (ECR/2017/001863).


  1. Alian, A.R., Kundalwal, S.I., Meguid, S.A.: Interfacial and mechanical properties of epoxy nanocomposites using different multiscale modeling schemes. Compos. Struct. 131, 545–555 (2015a). Google Scholar
  2. Alian, A.R., Kundalwal, S.I., Meguid, S.A.: Multiscale modeling of carbon nanotube epoxy composites. Polymer (U.K.) 70, 149–160 (2015b). Google Scholar
  3. Alian, A.R., Meguid, S.A., Kundalwal, S.I.: Unraveling the influence of grain boundaries on the mechanical properties of polycrystalline carbon nanotubes. Carbon 125, 180–188 (2017). Google Scholar
  4. Bahamon, D.A., Qi, Z., Park, H.S., Pereira, V.M., Campbell, D.K.: Conductance signatures of electron confinement induced by strained nanobubbles in graphene. Nanoscale 7(37), 15300–15309 (2015). Google Scholar
  5. Balandin, A.A., Ghosh, S., Bao, W., Calizo, I., Teweldebrhan, D., Miao, F., Lau, C.N.: Superior thermal conductivity of single-layer graphene. Nano Lett. 8, 902–907 (2008). Google Scholar
  6. Bastwros, M., Kim, G.Y.: Fabrication of custom pattern reinforced AZ31 multilayer composite using ultrasonic spray deposition. In: ASME 2016 11th International Manufacturing Science and Engineering Conference, MSEC-2016, vol. 1, pp. V001T02A016–V001T02A016 (2016).
  7. Benveniste, Y., Dvorak, G.J.: Uniform fields and universal relations in piezoelectric composites. J. Mech. Phys. Solids 40(6), 1295–1312 (1992). MathSciNetzbMATHGoogle Scholar
  8. Bernholc, J., Nakhmanson, S.M., Nardelli, M.B., Meunier, V.: Understanding and enhancing polarization in complex materials. Comput. Sci. Eng. 6(6), 12–21 (2004). Google Scholar
  9. Bhavanasi, V., Kumar, V., Parida, K., Wang, J., Lee, P.S.: Enhanced piezoelectric energy harvesting performance of flexible PVDF-TrFE bilayer films with graphene oxide. ACS Appl. Mater. Interfaces 8(1), 521–529 (2016). Google Scholar
  10. Chen, F., Gupta, N., Behera, R.K., Rohatgi, P.K.: Graphene-reinforced aluminum matrix composites: a review of synthesis methods and properties. JOM 70(6), 837–845 (2018). Google Scholar
  11. Conley, H., Lavrik, N.V., Prasai, D., Bolotin, K.I.: Graphene bimetallic-like cantilevers: probing graphene/substrate interactions. Nano Lett. 11(11), 4748–4752 (2011). Google Scholar
  12. Cui, Y., Kundalwal, S.I., Kumar, S.: Gas barrier performance of graphene/polymer nanocomposites. Carbon 98, 313–333 (2016). Google Scholar
  13. Da Cunha Rodrigues, G., Zelenovskiy, P., Romanyuk, K., Luchkin, S., Kopelevich, Y., Kholkin, A.: Strong piezoelectricity in single-layer graphene deposited on SiO2 grating substrates. Nat. Commun. 6, 7572 (2015). Google Scholar
  14. Dasari, B.L., Morshed, M., Nouri, J.M., Brabazon, D., Naher, S.: Mechanical properties of graphene oxide reinforced aluminium matrix composites. Compos. B Eng. 145, 136–144 (2018). Google Scholar
  15. Dewapriya, M.A.N., Rajapakse, R.K.N.D., Nigam, N.: Influence of hydrogen functionalization on the fracture strength of graphene and the interfacial properties of graphene-polymer nanocomposite. Carbon 93, 830–842 (2015). Google Scholar
  16. Gao, X.L., Li, K.: A shear-lag model for carbon nanotube-reinforced polymer composites. Int. J. Solids Struct. 42(5–6), 1649–1667 (2005). zbMATHGoogle Scholar
  17. García-Macías, E., Rodríguez-Tembleque, L., Sáez, A.: Bending and free vibration analysis of functionally graded graphene vs. carbon nanotube reinforced composite plates. Compos. Struct. 186, 123–138 (2018). Google Scholar
  18. Gharbi, M., Sun, Z.H., Sharma, P., White, K., El-Borgi, S.: Flexoelectric properties of ferroelectrics and the nanoindentation size-effect. Int. J. Solids Struct. 48(2), 249–256 (2011). zbMATHGoogle Scholar
  19. Gradinar, D.A., Mucha-Kruczyński, M., Schomerus, H., Fal’Ko, V.I.: Transport signatures of pseudomagnetic landau levels in strained graphene ribbons. Phys. Rev. Lett. 110(26), 266801 (2013). Google Scholar
  20. Gupta, S.S., Batra, R.C.: Elastic properties and frequencies of free vibrations of single-layer graphene sheets. J. Comput. Theor. Nanosci. 7(10), 2151–2164 (2010). Google Scholar
  21. Hadjesfandiari, A.R.: Size-dependent piezoelectricity. Int. J. Solids Struct. 50(18), 2781–2791 (2013). Google Scholar
  22. Hwang, S.H., Park, H.W., Park, Y.B.: Piezoresistive behavior and multi-directional strain sensing ability of carbon nanotube-graphene nanoplatelet hybrid sheets. Smart Mater. Struct. 22(1), 015013 (2013). Google Scholar
  23. Ji, X., Xu, Y., Zhang, W., Cui, L., Liu, J.: Review of functionalization, structure and properties of graphene/polymer composite fibers. Compos. A Appl. Sci. Manuf. 87, 29–45 (2016). Google Scholar
  24. Jiang, B., Liu, C., Zhang, C., Liang, R., Wang, B.: Maximum nanotube volume fraction and its effect on overall elastic properties of nanotube-reinforced composites. Compos. B Eng. 40(3), 212–217 (2009). Google Scholar
  25. Kandpal, M., Palaparthy, V., Tiwary, N., Rao, V.R.: Low cost, large area, flexible graphene nanocomposite films for energy harvesting applications. IEEE Trans. Nanotechnol. 16(2), 259–264 (2017). Google Scholar
  26. Khan, U., Young, K., O’Neill, A., Coleman, J.N.: High strength composite fibres frompolyester filled with nanotubes and graphene. J. Mater. Chem. 22(25), 12907–12914 (2012). Google Scholar
  27. Kothari, R., Kundalwal, S. I., Sahu, S.K.: Transversely isotropic thermal properties of carbon nanotubes containing vacancies. Acta Mech. 229, 2787–2800 (2018). Google Scholar
  28. Kumar, A., Chakraborty, D.: Effective properties of thermo-electro-mechanically coupled piezoelectric fiber reinforced composites. Mater. Des. 30, 1216–1222 (2009). Google Scholar
  29. Kundalwal, S.I.: Review on micromechanics of nano- and micro- fiber reinforced composites. Polym Compos (2017). Google Scholar
  30. Kundalwal, S.I., Choyal, V.: Transversely isotropic elastic properties of carbon nanotubes containing vacancy defects using MD. Acta Mech. 229, 2571–2584 (2018). Google Scholar
  31. Kundalwal, S.I., Meguid, S.A.: Multiscale modeling of regularly staggered carbon fibers embedded in nano-reinforced composites. Eur. J. Mech. A/Solids 64, 69–84 (2017). MathSciNetzbMATHGoogle Scholar
  32. Kundalwal, S.I., Ray, M.C.: Micromechanical analysis of fuzzy fiber reinforced composites. Int. J. Mech. Mater. Des. 7(2), 149–166 (2011). Google Scholar
  33. Kundalwal, S.I., Ray, M.C.: Effective properties of a novel composite reinforced with short carbon fibers and radially aligned carbon nanotubes. Mech. Mater. 53, 47–60 (2012). Google Scholar
  34. Kundalwal, S.I., Ray, M.C.: Effect of carbon nanotube waviness on the elastic properties of the fuzzy fiber reinforced composites. J. Appl. Mech. 80(2), 21010 (2013). Google Scholar
  35. Kundalwal, S.I., Meguid, S.A., Weng, G.J.: Strain gradient polarization in graphene. Carbon 117, 462–472 (2017). Google Scholar
  36. Kundalwal, S.I., Suresh Kumar, R., Ray, M.C.: Smart damping of laminated fuzzy fiber reinforced composite shells using 1-3 piezoelectric composites. Smart Mater. Struct. 22(10), 105001 (2013). Google Scholar
  37. Kvashnin, A.G., Sorokin, P.B., Yakobson, B.I.: Flexoelectricity in carbon nanostructures: nanotubes, fullerenes, and nanocones. J. Phys. Chem. Lett. 6(14), 2740–2744 (2015). Google Scholar
  38. Lee, C., Wei, X., Kysar, J.W., Hone, J.: Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321(5887), 385–388 (2008). Google Scholar
  39. Li, P., You, Z., Cui, T.: Graphene cantilever beams for nano switches. Appl. Phys. Lett. 101(9), 093111 (2012). Google Scholar
  40. Li, A., Zhou, S., Zhou, S., Wang, B.: Size-dependent analysis of a three-layer microbeam including electromechanical coupling. Compos. Struct. 116(1), 120–127 (2014). Google Scholar
  41. Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4(6), 637–642 (1968). zbMATHGoogle Scholar
  42. Moreno, M.E., Tita, V., Marques, F.D.: Finite element analysis applied to evaluation of effective material coefficients for piezoelectric fiber composites. In: Brazilian Symposium on Aerospace Eng. & Applications, 2005 (2009)Google Scholar
  43. Morozovska, A.N., Eliseev, E.A., Tagantsev, A.K., Bravina, S.L., Chen, L.Q., Kalinin, S.V.: Thermodynamics of electromechanically coupled mixed ionic-electronic conductors: deformation potential, Vegard strains, and flexoelectric effect. Phys. Rev. B Condens. Matter Mater. Phys. 83(19), 195313 (2011). Google Scholar
  44. Muñoz-Hernández, A., Diaz, G., Calderón-Muñoz, W.R., Leal-Quiros, E.: Thermal-electric modeling of graphite: analysis of charge carrier densities and Joule heating of intrinsic graphite rods. J. Appl. Phys. 122(24), 245107 (2017). Google Scholar
  45. Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., Grigorieva, I.V., Firsov, A.A.: Electric field effect in atomically thin carbon films. Science (New York, N.Y.) 306(5696), 666–669 (2004). Google Scholar
  46. Odegard, G.M.: Constitutive modeling of piezoelectric polymer composites Constitutive modeling of piezoelectric polymer composites constitutive modeling of piezoelectric polymer composites. Acta Mater. 52(18), 5315–5330 (2004)Google Scholar
  47. Park, J.Y., Park, C.H., Park, J.S., Kong, K.J., Chang, H., Im, S.: Multiscale computations for carbon nanotubes based on a hybrid QM/QC (quantum mechanical and quasicontinuum) approach. J. Mech. Phys. Solids 58(2), 86–102 (2010)Google Scholar
  48. Pettermann, H.E., Suresh, S.: A comprehensive unit cell model: a study of coupled effects in piezoelectric 1-3 composites. Int. J. Solids Struct. 37(39), 5447–5464 (2000). zbMATHGoogle Scholar
  49. Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117(1), 1–19 (1995). zbMATHGoogle Scholar
  50. Ray, M.C., Pradhan, A.K.: The performance of vertically reinforced 1-3 piezoelectric composites in active damping of smart structures. Smart Mater. Struct. 15(2), 631–641 (2006). Google Scholar
  51. Roberts, M.W., Clemons, C.B., Wilber, J.P., Young, G.W., Buldum, A., Quinn, D.D.: Continuum plate theory and atomistic modeling to find the flexural rigidity of a graphene sheet interacting with a substrate. J. Nanotechnol. 2010, 1–8 (2010)Google Scholar
  52. Rupa, N.S., Ray, M.C.: Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity. Int. J. Mech. Mater. Des. 13(3), 453–467 (2017). Google Scholar
  53. Saber, N., Araby, S., Meng, Q., Hsu, H.-Y., Yan, C., Azari, S., Lee, S.-H., Xu, Y., Ma, J., Yu, S.: Superior piezoelectric composite films: taking advantage of carbon nanomaterials. Nanotechnology 25(4), 045501 (2014). Google Scholar
  54. Scari, A.S., Pockszevnicki, B.C., Junior, J.L., Junior, P.A.A.M.: Stress-strain compression of AA6082-T6 aluminum alloy at room temperature. J. Struct. (2014). Google Scholar
  55. Shah, P. H., Batra, R. C.: Elastic moduli of covalently functionalized single layer graphene sheets. Comput. Mater. Sci. 95, 637–650 (2014a). Google Scholar
  56. Shah, P. H., Batra, R. C.: In-plane elastic moduli of covalently functionalized single-wall carbon nanotubes. Comput. Mater. Sci. 83, 349–361 (2014b). Google Scholar
  57. Shen, S., Hu, S.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58(5), 665–677 (2010). MathSciNetzbMATHGoogle Scholar
  58. Smith, W.A., Auld, B.A.: Modeling 1–3 composite piezoelectrics: thickness-mode oscillations. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38(1), 40–47 (1991). Google Scholar
  59. Song, Y.S., Youn, J.R.: Modeling of effective elastic properties for polymer based carbon nanotube composites. Polymer 47(5), 1741–1748 (2006). Google Scholar
  60. Stuart, S.J., Tutein, A.B., Harrison, J.A.: A reactive potential for hydrocarbons with intermolecular interactions. J. Chem. Phys. 112(14), 6472 (2000). Google Scholar
  61. Sundar, U., Cook-Chennault, K.A., Banerjee, S., Refour, E.: Dielectric and piezoelectric properties of percolative three-phase piezoelectric polymer composites. J. Vac. Sci. Technol. B, Nanotechnol. Microelectron. Mater. Process. Meas. Phenom. 34(4), 41232 (2016). Google Scholar
  62. Tian, W., Li, S., Wang, B., Chen, X., Liu, J., Yu, M.: Graphene-reinforced aluminum matrix composites prepared by spark plasma sintering. Int. J. Miner. Metall. Mater. 23(6), 723–729 (2016). Google Scholar
  63. Verma, D., Gupta, S.S., Batra, R.C.: Vibration mode localization in single- and multi-layered graphene nanoribbons. Comput. Mater. Sci. 95, 41–52 (2014). Google Scholar
  64. Wang, J., Li, Z., Fan, G., Pan, H., Chen, Z., Zhang, D.: Reinforcement with graphene nanosheets in aluminum matrix composites. Scr. Mater. 66(8), 594–597 (2012). Google Scholar
  65. Yan, Z., Jiang, L.Y.: Flexoelectric effect on the electroelastic responses of bending piezoelectric nanobeams flexoelectric effect on the electroelastic responses of bending piezoelectric nanobeams. J. Appl. Phys. 113(19), 194102 (2013). Google Scholar
  66. Ying, C., Zhifei, S.: Exact solutions of functionally gradient piezothermoelastic cantilevers and parameter identification. J. Intell. Mater. Syst. Struct. 16(6), 531–539 (2005). Google Scholar
  67. Zhang, Y.B., Tan, Y.W., Stormer, H.L., Kim, P.: Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438(7065), 201–204 (2005). Google Scholar
  68. Zhang, Y.Y., Pei, Q.X., Wang, C.M.: Mechanical properties of graphynes under tension: a molecular dynamics study. Appl. Phys. Lett. 101(8), 081909 (2012). Google Scholar
  69. Zhao, X., Zhang, Q., Chen, D., Lu, P.: Enhanced mechanical properties of graphene-based poly(vinyl alcohol) composites. Macromolecules 43, 2357–2363 (2010)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • S. I. Kundalwal
    • 1
    Email author
  • K. B. Shingare
    • 1
  • Ankit Rathi
    • 1
  1. 1.Applied and Theoretical Mechanics (ATOM) Laboratory, Discipline of Mechanical EngineeringIndian Institute of Technology IndoreIndoreIndia

Personalised recommendations