Advertisement

Acta Mechanica

, Volume 230, Issue 10, pp 3613–3632 | Cite as

Maxwell homogenization scheme for piezoelectric composites with arbitrarily-oriented spheroidal inhomogeneities

  • R. Rodríguez-RamosEmail author
  • C. A. Gandarilla-Pérez
  • L. Lau-Alfonso
  • F. Lebon
  • F. J. Sabina
  • I. Sevostianov
Original Paper
  • 51 Downloads

Abstract

In this work, the effective electro-elastic properties of piezoelectric composites are computed using the Maxwell homogenization method (MHM). The composites are made by several families of spheroidal inhomogeneities embedded in a homogeneous infinite medium (matrix). Each family of spheroidal inhomogeneities is made of the same material, and all the inhomogeneities have identical size and shape and are randomly oriented. The inhomogeneities and matrix materials exhibit piezoelectric transversely isotropic symmetry. It is shown that the shape of the “effective inclusion” substantially affects the effective piezoelectric properties. A new and simple form to calculate the aspect ratio of effective inclusion is presented. The effect on the overall piezoelectric properties due to the orientation of the inhomogeneities and different families of piezoelectric inhomogeneities is discussed. The MHM approach is applied in two examples, material with inhomogeneities having scatter orientation and composites with two different families of spheroidal inhomogeneities.

Notes

Acknowledgements

The funding of Proyecto Nacional de Ciencias Básicas 2013-2015 (Project No. 7515) is gratefully acknowledged. Thanks to the Mathematics and Mechanics Department at IIMAS-UNAM and FENOMEC for their support and to Ramiro Chávez Tovar and Ana Pérez Arteaga for computational assistance. The authors would like to thank the project PHC Carlos J. Finlay 2018 Project No. 39142TA (France–Cuba) and the French embassy in Havana for their support on travel expenses of PhD students in 2018. The author Rodríguez-Ramos would like to thank MyM-IIMAS-UNAM and PREI-DGAPA-UNAM for the financial support provided.

References

  1. 1.
    Maxwell, J.: A Treatise on Electricity and Magnetism. Clarendon Press, Oxford (1873)zbMATHGoogle Scholar
  2. 2.
    Kuster, G., Toksöz, M.N.: Velocity and attenuation of seismic waves in two-phase media I. Theoretical formulations. Geophysics 39, 587–606 (1974)CrossRefGoogle Scholar
  3. 3.
    Shen, L., Yi, S.: An effective inclusion model for effective moduli of heterogeneous materials with ellipsoidal inhomogeneities. Int. J. Solids Struct. 38, 5789–5805 (2001)CrossRefGoogle Scholar
  4. 4.
    McCartney, L., Kelly, A.: Maxwell’s far-field methodology applied to the prediction of properties of multi-phase isotropic particulate composites. Proc. R. Soc. Lond. A 464, 423–446 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    McCartney, L.: Maxwell’s far-field methodology predicting elastic properties of multiphase composites reinforced with aligned transversely isotropic spheroids. Philos. Mag. 90, 4175–4207 (2010)CrossRefGoogle Scholar
  6. 6.
    Sevostianov, I., Giraud, A.: Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape. Int. J. Eng. Sci. 64, 23–36 (2013)CrossRefGoogle Scholar
  7. 7.
    Sevostianov, I.: On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites. Mech. Mater. 75, 45–59 (2014)CrossRefGoogle Scholar
  8. 8.
    Kushch, V., Mogilevskaya, S., Stolarski, H., Crouch, S.: Evaluation of the effective elastic moduli of particulate composites based on Maxwell’s concept of equivalent inhomogeneity: microstructure-induced anisotropy. J. Mech. Mater. Struct. 8, 283–303 (2012)CrossRefGoogle Scholar
  9. 9.
    Vilchevskaya, E., Sevostianov, I.: Scattering and attenuation of elastic waves in random media. Int. J. Eng. Sci. 94, 139–149 (2015)CrossRefGoogle Scholar
  10. 10.
    Gandarilla-Pérez, C.A., Rodríguez-Ramos, R., Sevostianov, I., Sabina, F.J., Bravo-Castillero, J., Guinovart-Díaz, R., Lau-Alfonso, L.: Extension of Maxwell homogenization scheme for piezoelectric composites containing spheroidal inhomogeneities. Int. J. Solids Struct. 135, 125–136 (2017).  https://doi.org/10.1016/j.ijsolstr.2017.11.015 CrossRefGoogle Scholar
  11. 11.
    Li, J.Y., Dunn, M.L.: Variational bounds for the effective moduli of heterogeneous piezoelectric solids. Philos. Mag. A 81, 903–926 (2001)CrossRefGoogle Scholar
  12. 12.
    Min, C., Yu, D., Cao, J., Wang, G., Feng, L.: A graphite nanoplatelet/epoxy composite with high dielectric constant and high thermal conductivity. Carbon 55, 116–125 (2013)CrossRefGoogle Scholar
  13. 13.
    Wang, D., Zhang, X., Zha, J.-W., Zhao, J., Dang, Z.-M., Hu, G.-H.: Dielectric properties of reduced graphene oxide/polypropylene composites with ultralow percolation threshold. Polymer 54, 1916–1922 (2013)CrossRefGoogle Scholar
  14. 14.
    Yousefi, N., Sun, X., Lin, X., Shen, X., Jia, J., Zhang, B., Tang, B., Chan, M., Kim, J.-K.: Highly aligned graphene/polymer nanocomposites with excellent dielectric properties for high-performance electromagnetic interference shielding. Adv. Mater. 26, 5480–5487 (2014)CrossRefGoogle Scholar
  15. 15.
    Xia, X., Wang, Y., Zhong, Z., Weng, G.J.: A theory of electrical conductivity, dielectric constant, and electromagnetic interference shielding for lightweight graphene composite foams. J. Appl. Phys. 120, 085102 (2016)CrossRefGoogle Scholar
  16. 16.
    Xia, X., Mazzeo, A.D., Zhong, Z., Weng, G.J.: An X-band theory of electromagnetic interference shielding for graphene-polymer nanocomposites. J. Appl. Phys. 122, 025104 (2017)CrossRefGoogle Scholar
  17. 17.
    Xia, X., Wang, Y., Zhong, Z., Weng, G.J.: A frequency-dependent theory of electrical conductivity and dielectric permittivity for graphene-polymer nanocomposites. Carbon 111, 221–230 (2017)CrossRefGoogle Scholar
  18. 18.
    Weng, G.J.: A dynamical theory for the Mori–Tanaka and Ponte Castañeda–Willis estimates. Mech. Mater. 42(9), 886–893 (2010)CrossRefGoogle Scholar
  19. 19.
    Wang, Y., Weng, G.J., Meguid, S.A., Hamouda, A.M.: A continuum model with a percolation threshold and tunneling-assisted interfacial conductivity for carbon nanotube-based nanocomposites. J. Appl. Phys. 115(19), 193706 (2014)CrossRefGoogle Scholar
  20. 20.
    Wang, Y., Su, Y., Li, J., Weng, G.J.: A theory of magnetoelectric coupling with interface effects and aspect-ratio dependence in piezoelectric-piezomagnetic composites. J. Appl. Phys. 117(16), 164106 (2015)CrossRefGoogle Scholar
  21. 21.
    Kachanov, M., Sevostianov, I.: On quantitative characterization of microstructures and effective properties. Int. J. Solids Struct. 42, 309–336 (2005)CrossRefGoogle Scholar
  22. 22.
    Chou, T., Nomura, S.: Fibre orientation effects on the thermoelastic properties of short-fiber composites. Sci. Technol. 14, 279–291 (1981)Google Scholar
  23. 23.
    Takao, Y., Chou, T., Taya, M.: Effective longitudinal Young’s modulus of misoriented short fiber composites. J. Appl. Mech. 49, 536–540 (1982)CrossRefGoogle Scholar
  24. 24.
    Ferrari, M., Johnson, M.: Effective elasticities of short-fiber composites with arbitrary orientation distribution. Mech. Mater. 8, 67–73 (1989)CrossRefGoogle Scholar
  25. 25.
    Barnett, D., Lothe, J.: Dislocations and line charges in anisotropic piezoelectric insulators. Phys. Status Solidi B 67, 105–111 (1975)CrossRefGoogle Scholar
  26. 26.
    Levin, V.M., Michelitsch, T., Sevostianov, I.: Spheroidal inhomogeneity in the transversely isotropic piezoelectric medium. Arch. Appl. Mech. 70, 673–693 (2000)CrossRefGoogle Scholar
  27. 27.
    Rodríguez-Ramos, R., Gandarilla-Pérez, C., Otero, J.: Static effective characteristics in piezoelectric composite materials. Math. Methods Appl. Sci. 40, 3249–3264 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Dunn, M.: Electroelastic Green’s functions for transversely isotropic piezoelectric media and their application to the solution of inclusion and inhomogeneity problems. Int. J. Eng. Sci. 32, 119–131 (1994)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lu, Y., Liaw, P.: Effect of particle orientation in silicon-carbide particle-reinforced aluminium-matrix composite extrusions on ultrasonic velocity-measurements. J. Compos. Mater. 29, 1096–1116 (1995)CrossRefGoogle Scholar
  30. 30.
    Chen, C., Wang, Y.: Effective thermal conductivity of misoriented short fiber reinforced thermoplastics. Mech. Mater. 23, 217–228 (1996)CrossRefGoogle Scholar
  31. 31.
    Pettermann, H., Böhm, H., Rammerstorfer, F.: Some direction dependent properties of matrix-inclusion type composites with given reinforcement orientation distributions. Compos. Part B Eng. 28, 253–265 (1997)CrossRefGoogle Scholar
  32. 32.
    Fu, S., Lauke, B.: The elastic modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 58, 389–400 (1998)CrossRefGoogle Scholar
  33. 33.
    Sevostianov, I., Kachanov, M.: Modeling of the anisotropic elastic properties of plasma-sprayed coatings in relation to their microstructure. Acta Mater. 48, 1361–1370 (2000)CrossRefGoogle Scholar
  34. 34.
    Sevostianov, I., Levin, V., Radi, E.: Effective viscoelastic properties of short-fiber reinforced composites. Int. J. Eng. Sci. 100, 61–73 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mishurova, T., Rachmatulin, N., Fontana, P., Oesch, T., Bruno, G., Radi, E., Sevostianov, I.: Evaluation of the probability density of inhomogeneous fiber orientations by computed tomography and its application to the calculation of the effective properties of a fiber-reinforced composite. Int. J. Eng. Sci. 122, 14–29 (2018)CrossRefGoogle Scholar
  36. 36.
    Giraud, A., Huynh, Q., Hoxha, D., Kondo, D.: Effective poroelastic properties of transversely isotropic rock-like composites with arbitrarily oriented ellipsoidal inclusions. Mech. Mater. 39, 1006–1024 (2007)CrossRefGoogle Scholar
  37. 37.
    Kachanov, M., Tsukrov, I., Shafiro, B.: Effective properties of solids with randomly located defects. In: Breusse, D. (ed.) Probabilities and Materials: Tests Models and Applications, pp. 225–240. Kluwer Publications, Dordrecht (1994)CrossRefGoogle Scholar
  38. 38.
    Levin, V.: The effective properties of piezoactive matrix composite materials. J. Appl. Math. Mech. 60(2), 309–317 (1996)CrossRefGoogle Scholar
  39. 39.
    Berlincourt, D.A.: Piezoelectric Crystals and Ceramics. Ultrasonic Transducer Materials. Springer, Boston (1971)Google Scholar
  40. 40.
    Chan, H., Unsworth, J.: Simple model for piezoelectric ceramic/polymer 1–3 composites used in ultrasonic transducer applications. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 36(4), 434 (1989)CrossRefGoogle Scholar
  41. 41.
    Kar-Gupta, R., Venkatesh, T.: Electromechanical response of 1–3 piezoelectric composite: effect of poling characteristics. J. Appl. Phys. 98, 054102 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Facultad de Matemática y ComputaciónUniversidad de La HabanaHavanaCuba
  2. 2.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoMexicoMexico
  3. 3.Facultad de FísicaUniversidad de La HabanaHavanaCuba
  4. 4.Instituto de Cibernética Matemática y FísicaHavanaCuba
  5. 5.CNRS, Centrale Marseille, LMAAix-Marseille UniversityMarseille Cedex 13France
  6. 6.Department of Mechanical and Aerospace EngineeringNew Mexico State UniversityLas CrucesUSA

Personalised recommendations