Applied Mathematics and Mechanics

, Volume 39, Issue 6, pp 797–812 | Cite as

Effective electroelastic constants for three-phase confocal elliptical cylinder model in piezoelectric quasicrystal composites

  • Yongbin Wang
  • Junhong Guo


A three-phase confocal elliptical cylinder model is proposed to analyze micromechanics of one-dimensional hexagonal piezoelectric quasicrystal (PQC) composites. Exact solutions of the phonon, phason, and electric fields are obtained by using the conformal mapping combined with the Laurent expansion technique when the model is subject to far-field anti-plane mechanical and in-plane electric loadings. The effective electroelastic constants of several different composites made up of PQC, quasicrystal (QC), and piezoelectric (PE) materials are predicted by the generalized self-consistent method. Numerical examples are conducted to show the effects of the volume fraction and the cross-sectional shape of inclusion (or fiber) on the effective electroelastic constants of these composites. Compared with other micromechanical methods, the generalized selfconsistent and Mori-Tanaka methods can predict the effective electroelastic constants of the composites consistently.

Key words

piezoelectric quasicrystal (PQC) three-phase elliptical cylinder model effective constant generalized self-consistent method 

Chinese Library Classification


2010 Mathematics Subject Classification

52C23 74A60 74B05 74F99 74G05 


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  1. [1]
    Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53, 1951–1953 (1984)CrossRefGoogle Scholar
  2. [2]
    Fujiwara, T., Laissardiere, G. T., and Yamamoto, S. Electronic structure and electron transport in quasicrystals. Materials Science Forum, 150–151, 387–394 (1994)CrossRefGoogle Scholar
  3. [3]
    Ding, D. H., Yang, W. G., Hu, C. Z., and Wang, R. Generalized elasticity theory of quasicrystals. Physical Review B, 48, 7003–7009 (1993)CrossRefGoogle Scholar
  4. [4]
    Yang, W. G., Wang, R. H., Ding, D. H., and Hu, C. Linear elasticity theory of cubic quasicrystals. Physical Review B, 48, 6999–7002 (1993)CrossRefGoogle Scholar
  5. [5]
    Hu, C., Wang, R., Yang, W., and Ding, D. Point groups and elastic properties of two-dimensional quasicrystals. Acta Crystallographica, 52, 251–256 (1996)CrossRefGoogle Scholar
  6. [6]
    Hu, C. Z., Wang, R., and Ding, D. H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Reports on Progress in Physics, 63, 1–1 (2000)MathSciNetCrossRefGoogle Scholar
  7. [7]
    Fan, T. Y. and Mai, Y. W. Elasticity theory fracture mechanics and some relevant thermal properties of quasi-crystalline materials. Applied Mechanics Reviews, 57, 325–343 (2004)CrossRefGoogle Scholar
  8. [8]
    Fan, T. Y. and Guo, L. H. The final governing equation and fundamental solution of plane elasticity of icosahedral quasicrystals. Physics Letters A, 341, 235–239 (2005)CrossRefzbMATHGoogle Scholar
  9. [9]
    Thiel, P. A. and Dubois, J. M. Quasicrystals reaching maturity for technological applications. Materials Today, 2, 3–7 (1999)CrossRefGoogle Scholar
  10. [10]
    Park, J. Y., Sacha, G. M, Enachescu, M., Ogletree, D. F., Ribeiro, R. A., Canfield, P. C., Genks, C. J., Thiel, P. A., Saenz, J. J., and Salmeron, M. Sensing dipole fields at atomic steps with combined scanning tunneling and force microscopy. Physical Review Letters, 95, 136802 (2005)CrossRefGoogle Scholar
  11. [11]
    Engel, M., Umezaki, M., Trebin, H. R., and Odagaki, T. Dynamics of particle flips in twodimensional quasicrystals. Physical Review B Condensed Matter, 82, 087201 (2010)CrossRefGoogle Scholar
  12. [12]
    Sakly, A., Kenzari, S., Bonina, D., Corbel, S., Fournee, V. A novel quasicrystal-resin composite for stereolithography. Materials and Design, 56, 280–285 (2014)CrossRefGoogle Scholar
  13. [13]
    Guo, X. P., Chen, J. F., Yu, H. L., Liao, H., and Coddet, C. A study on the microstructure and tribological behavior of cold-sprayed metal matrix composites reinforced by particulate quasicrystal. Surface and Coatings Technology, 268, 94–98 (2015)CrossRefGoogle Scholar
  14. [14]
    Zhang, Y., Zhang, J., Wu, G. H., Liu, W. C., Zhang, L., and Ding, W. J. Microstructure and tensile properties of as-extruded Mg-Li-Zn-Gd alloys reinforced with icosahedral quasicrystal phase. Materials and Design, 66, 162–168 (2015)CrossRefGoogle Scholar
  15. [15]
    Tian, Y., Huang, H., Yuan, G. Y., Chen, C. L., Wang, Z. C., and Ding, W. J. Nanoscale icosahedral quasicrystal phase precipitation mechanism during annealing for Mg-Zn-Gd based alloys. Materials Letters, 130, 236–239 (2014)CrossRefGoogle Scholar
  16. [16]
    Zhang, D. L. Electronic properties of stable decagonal quasicrystals. Physical Status Solidi A, 207, 2666–2673 (2010)CrossRefGoogle Scholar
  17. [17]
    Yang, W. G., Wang, R., Ding, D. H., and Hu, C. Elastic strains induced by electric fields in quasicrystals. Physics Condensed Matter, 7, L499–L502 (1995)CrossRefGoogle Scholar
  18. [18]
    Li, C. L. and Liu, Y. Y. The physical property tensors of one dimensional quasicrystals. Chinese Physics B, 13, 924–931 (2004)CrossRefGoogle Scholar
  19. [19]
    Rao, K. R. M., Rao, P. H., and Chaitanya, B. S. K. Piezoelectricity in quasicrystals: a grouptheoretical study. Pramana—Journal of Physic, 68, 481–487 (2007)Google Scholar
  20. [20]
    Altay, G. and Dökmeci, M. C. On the fundamental equations of piezoelasticity of quasicrystal media. International Journal of Solids and Structures, 49, 3255–3262 (2012)CrossRefGoogle Scholar
  21. [21]
    Grimmer, H. The piezoelectric effect of second order in stress or strain: its form for crystals and quasicrystals of any symmetry. Acta Crystallographica, 63, 441–446 (2007)CrossRefGoogle Scholar
  22. [22]
    Zhang, L. L., Zhang, Y. T., and Gao, Y. General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect. Physics Letters A, 378, 2768–2776 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Yang, J. and Li, X. Analytic solutions of problem about a circular hole with a straight crack in one-dimensional hexagonal quasicrystals with piezoelectric effects. Theoretical Applied Fracture Mechanics, 82, 17–24 (2015)CrossRefGoogle Scholar
  24. [24]
    Li, X.Y., Li, P. D., Wu, T. H., Shi, M. X., and Zhu, Z. W. Three dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect. Physics Letters A, 378, 826–834 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Yu, J., Guo, J. H., Pan, E., and Xing, Y. General solutions of one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics. Applied Mathematics and Mechanics (English Edition), 36(6), 793–814 (2015) Scholar
  26. [26]
    Yu, J., Guo, J. H., and Xing, Y. M. Complex variable method for an anti-plane elliptical cavity of one-dimensional hexagonal piezoelectric quasicrystals. Chinese Journal of Aeronautics, 28, 1287–1295 (2015)CrossRefGoogle Scholar
  27. [27]
    Zhang, L. L., Wu, D., Xu, W. S., Yang, L. Z., Ricoeur, A., Wang, Z. B., and Gao, Y. Green’s functions of one-dimensional quasicrystal bi-material with piezoelectric effect. Physics Letters A, 380, 3222–3228 (2016)MathSciNetCrossRefGoogle Scholar
  28. [28]
    Dunn, M. L. and Taya, M. Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures, 30, 161–175 (1993)CrossRefzbMATHGoogle Scholar
  29. [29]
    Christensen, R. M. and Lo, K. H. Solutions for effective shear properties in three phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids, 27, 315–330 (1979)CrossRefzbMATHGoogle Scholar
  30. [30]
    Huang, Y., Hu, K. X., Wei, X., and Chandra, A. A generalized self-consistent mechanics method for composite materials with multiphase inclusion. Journal of the Mechanics and Physics of Solids, 42, 491–504 (1994)CrossRefzbMATHGoogle Scholar
  31. [31]
    Jiang, C. P., Tong, Z. H., and Cheung, Y. K. A generalized self-consistent method for piezoelectric fiber reinforced composites under antiplane shear. Mechanics of Materials, 33, 295–308 (2001)CrossRefGoogle Scholar
  32. [32]
    Guo, J. H., Zhang, Z. Y., and Xing, Y. M. Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites. Philosophical Magazine, 96, 349–369 (2016)CrossRefGoogle Scholar
  33. [33]
    Guo, J. H. and Pan, E. Three-phase cylinder model of one-dimensional hexagonal piezoelectric quasi-crystal composites. Journal of Applied Mechanics, 83, 081007 (2016)CrossRefGoogle Scholar
  34. [34]
    Chen, F. L. and Jiang, C. P. A three-phase confocal elliptical cylinder model for predicting the thermal conductivity of composites. Proceedings of SPIE—The International Society for Optical Engineering, 74936 (2009)Google Scholar
  35. [35]
    Xiao, J. H., Xu, Y. L., and Zhang, F. C. A generalized self-consistent method for nano composites accounting for fiber section shape under antiplane shear. Mechanics of Materials, 81, 94–100 (2015)CrossRefGoogle Scholar
  36. [36]
    Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity, Springer, Holland (1953)zbMATHGoogle Scholar
  37. [37]
    Jiang, C. P. and Cheung, Y. K. A fiber/matrix/composite model with a combined conformal elliptical cylinder unit cell for predicting the effective longitudinal shear modulus. International Journal of Solids and Structures, 35, 3977–3987 (1998)CrossRefzbMATHGoogle Scholar
  38. [38]
    Whitney, J. M. and Riley, M. B. Elastic properties of fiber reinforced composite materials. AIAA Journal, 4, 1537–1542 (1966)CrossRefGoogle Scholar
  39. [39]
    Shari, H. Z. and Chou, T. W. Transverse elastic moduli of unidirectional fiber composites with fiber/matrix interfacial debonding. Metallurgical Transactions A, 53, 383–391 (1995)Google Scholar
  40. [40]
    Fang, X. Q., Huang, M. J., Liu, J. X., and Feng, W. J. Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers. Composites Science and Technology, 98, 79–85 (2014)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MechanicsInner Mongolia University of TechnologyHohhotChina

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