Acta Mechanica

, Volume 230, Issue 5, pp 1781–1810 | Cite as

Nonlocal analytical solution of functionally graded multilayered one-dimensional hexagonal piezoelectric quasicrystal nanoplates

  • Li Zhang
  • Junhong GuoEmail author
  • Yongming Xing
Original Paper


Based on the nonlocal elasticity theory, the static bending deformation of a functionally graded multilayered one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) simply supported nanoplate is investigated under surface mechanical loadings. The functionally graded material is assumed to be exponential along the thickness direction. By utilizing the pseudo-Stroh formalism and propagator matrix method, exact closed-form solutions of functionally graded multilayered 1D hexagonal PQC nanoplates are then obtained by assuming that the layer interfaces are perfectly contacted. Numerical examples for six kinds of sandwich functionally graded nanoplates made up of piezoelectric crystals, quasicrystal and PQC are presented to illustrate the influence of the exponential factor, nonlocal parameter and stacking sequence on the phonon, phason and electric fields, which play an important role in designing new composite structures in engineering.



This work was supported by the National Natural Science Foundation of China (Grant Nos. 11862021, 11502123, 11262012, 11762013) and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2015JQ01).


  1. 1.
    Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53(20), 1951–1953 (1984)CrossRefGoogle Scholar
  2. 2.
    Levine, D., Steinhardt, P.J.: Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53(26), 2477–2480 (1984)CrossRefGoogle Scholar
  3. 3.
    Ding, D.H., Yang, W.G., Hu, C.Z., Wang, R.H.: Generalized elasticity theory of quasicrystals. Phys. Rev. B 48(10), 7003–7010 (1993)CrossRefGoogle Scholar
  4. 4.
    Balbyshev, V.N., King, D.J., Khramov, A.N., Kasten, L.S., Donley, M.S.: Investigation of quaternary Al-based quasicrystal thin films for corrosion protection. Thin Solid Films 447(3), 558–563 (2004)CrossRefGoogle Scholar
  5. 5.
    Thiel, P.A., Dubois, J.M.: Quasicrystals: reaching maturity for technological applications. Mater. Today 2(3), 3–7 (1999)CrossRefGoogle Scholar
  6. 6.
    Fujiwara, T.: Electronic structures and transport properties in quasicrystals. J. Non-Cryst. Solids 156–158(5), 865–871 (1993)CrossRefGoogle Scholar
  7. 7.
    Zhang, D.L.: Electronic properties of stable decagonal quasicrystals. Phys. Status Solidi A 207(12), 2666–2673 (2010)CrossRefGoogle Scholar
  8. 8.
    Yang, W.G., Wang, R.H., Ding, D.H., Hu, C.Z.: Elastic strains induced by electric fields in quasicrystals. J. Phys. Condens. Matter. 7(39), L499–L502 (1995)CrossRefGoogle Scholar
  9. 9.
    Li, C.L., Liu, Y.Y.: The physical property tensors of one-dimensional quasicrystals. Chin. Phys. 13(6), 924–931 (2004)CrossRefGoogle Scholar
  10. 10.
    Rao, K.R.M., Rao, P.H., Chaitanya, B.S.K.: Piezoelectricity in quasicrystals: a group-theoretical study. Pramana J. Phys. 68(3), 481–487 (2007)CrossRefGoogle Scholar
  11. 11.
    Altay, G., Dökmeci, M.C.: On the fundamental equations of piezoelasticity of quasicrystal media. Int. J. Solids Struct. 49(23–24), 3255–3262 (2012)CrossRefGoogle Scholar
  12. 12.
    Li, X.Y., Li, P.D., Wu, T.H., Shi, M.X., Zhu, Z.W.: Three dimensional fundamental solutions for one-dimensional hexagonal quasi-crystal with piezoelectric effect. Phys. Lett. A 378(10), 826–834 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Yu, J., Guo, J.H., Pan, E., Xing, Y.M.: General solutions of one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics. J. Appl. Math. Mech. 36(6), 793–814 (2015)zbMATHCrossRefGoogle Scholar
  14. 14.
    Yu, J., Guo, J.H., Xing, Y.M.: Complex variable method for an anti-plane elliptical cavity of one-dimensional hexagonal piezoelectric quasicrystals. Chin. J. Aeronaut. 28(4), 1287–1295 (2015)CrossRefGoogle Scholar
  15. 15.
    Guo, J.H., Zhang, Z.Y., Xing, Y.M.: Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites. Philos. Mag. 96(4), 349–369 (2016)CrossRefGoogle Scholar
  16. 16.
    Guo, J.H., Pan, E.: Three-phase cylinder model of one-dimensional hexagonal piezoelectric quasi-crystal composites. J. Appl. Mech. 83, 081007 (2016)CrossRefGoogle Scholar
  17. 17.
    Inoue, A., Takeuchi, A.: Recent progress in bulk glassy, nanoquasicrystalline and nanocrystalline alloys. Mater. Sci. Eng. A 375–377(1), 16–30 (2004)CrossRefGoogle Scholar
  18. 18.
    Wang, Z.F., Zhao, W.M., Qin, C.L., Cui, Y., Fan, S.L., Jia, J.J.: Direct preparation of nano-quasicrystals via a water-cooled wedge-shaped copper mould. J. Nanomater. 2012(1687–4110), 208–212 (2012)Google Scholar
  19. 19.
    Zhang, J.S., Pei, L.X., Du, H.W., Liang, W., Xu, C.H.X., Lu, B.F.: Effect of Mg-based spherical quasicrystals on microstructure and mechanical properties of AZ91 Alloys. J. Alloys Compd. 453, 309–315 (2008)CrossRefGoogle Scholar
  20. 20.
    Inoue, A., Kong, F., Zhu, S., Liu, C.T., Almarzouki, F.: Development and applications of highly functional Al-based materials by use of metastable phases. Mater. Res. 18, 1414–1425 (2015)CrossRefGoogle Scholar
  21. 21.
    Fournée, V., Sharma, H.R., Shimoda, M., Tsai, A.P., Unal, B., Ross, A.R., Lograsso, T.A., Thiel, P.A.: Quantum size effects in metal thin films grown on quasicrystalline substrates. Phys. Rev. Lett. 95(15), 155504 (2005)CrossRefGoogle Scholar
  22. 22.
    Lefaix, H., Prima, F., Zanna, S., Vermaut, P., Dubot, P., Marcus, P., Janickovic, D., Svec, P.: Surface properties of a nano-quasicrystalline forming Ti based system. Mater. Trans. 48(3), 278–286 (2007)CrossRefGoogle Scholar
  23. 23.
    Lam, D.C.C., Yang, F., Chong, A.C.M., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)zbMATHCrossRefGoogle Scholar
  24. 24.
    Akgöz, B., Civalek, Ö.: A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mech. 226(7), 2277–2294 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Guo, J.H., Chen, J.Y., Pan, E.: Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory. Physica E 87, 98–106 (2017)CrossRefGoogle Scholar
  26. 26.
    Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modelling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51(1), 303–313 (2014)CrossRefGoogle Scholar
  27. 27.
    Di, P.M., Failla, G., Pirrotta, A., Sofi, A., Zingales, M.: The mechanically based non-local elasticity: an overview of main results and future challenges. Proc. R. Soc. A 371, 20120433 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)CrossRefGoogle Scholar
  29. 29.
    Eringen, A.C.: Nonlocal Continuum Field Theories, vol. 46, pp. 391–398. Springer, New York (2002)zbMATHGoogle Scholar
  30. 30.
    Yan, J.W., Tong, L.H., Li, C., Zhu, Y., Wang, Z.W.: Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory. Compos. Struct. 125, 304–313 (2015)CrossRefGoogle Scholar
  31. 31.
    Sajadi, B., Goosen, H., Keulen, F.V.: Capturing the effect of thickness on size-dependent behavior of plates with nonlocal theory. Int. J. Solids Struct. 115–116, 140–148 (2017)CrossRefGoogle Scholar
  32. 32.
    Waksmanski, N., Pan, E.: Nonlocal analytical solutions for multilayered one-dimensional quasicrystal nanoplates. J. Vib. Acoust. 139(2), 021006 (2017)CrossRefGoogle Scholar
  33. 33.
    Zhang, L., Guo, J.H., Xing, Y.M.: Bending deformation of multilayered one-dimensional hexagonal piezoelectric quasicrystal nanoplates with nonlocal effect. Int. J. Solids Struct. 132–133, 278–302 (2018)CrossRefGoogle Scholar
  34. 34.
    Wei, D.X., He, Z.B.: Multilayered sandwich-like architecture containing large-scale faceted Al–Cu–Fe quasicrystal grains. Mater. Charact. 111, 154–161 (2016)CrossRefGoogle Scholar
  35. 35.
    Yang, L.Z., Gao, Y., Pan, E., Waksmanski, N.: An exact solution for a multilayered two-dimensional decagonal quasicrystal plate. Int. J. Solids Struct. 51(9), 1737–1749 (2014)CrossRefGoogle Scholar
  36. 36.
    Yang, L.Z., Gao, Y., Pan, E., Waksmanski, N.: An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate. Acta Mech. 226(11), 3611–3621 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Sladek, J., Sladek, V., Pan, E.: Bending analyses of 1D orthorhombic quasicrystal plates. Int. J. Solids Struct. 50(24), 3975–3983 (2013)CrossRefGoogle Scholar
  38. 38.
    Gao, Y., Xu, S.P., Zhao, B.S.: Boundary conditions for plate bending in one-dimensional hexagonal quasicrystals. J. Elast. 86(3), 221–233 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Waksmanski, N., Pan, E., Yang, L.Z., Gao, Y.: Free vibration of a multilayered one-dimensional quasi-crystal Plate. J. Vib. Acoust. 136(4), 63–69 (2014)CrossRefGoogle Scholar
  40. 40.
    Waksmanski, N., Pan, E., Yang, L.Z., Gao, Y.: Harmonic response of multilayered one-dimensional quasicrystal plates subjected to patch loading. J. Sound Vib. 375, 237–253 (2016)CrossRefGoogle Scholar
  41. 41.
    Erdogan, F.: Fracture mechanics of functionally graded materials. Compos. Eng. 5, 753–770 (1995)CrossRefGoogle Scholar
  42. 42.
    Suresh, S., Mortensen, A.: Fundamentals of functionally graded materials. Mater. Today 1(98), 18 (1998)Google Scholar
  43. 43.
    Suresh, S.: Graded materials for resistance to contact deformation and damage. Science 292(5526), 2447–2451 (2001)CrossRefGoogle Scholar
  44. 44.
    Pan, E., Han, F.: Exact solution for functionally graded and layered magneto-electro-elastic plates. Int. J. Eng. Sci. 43(3–4), 321–339 (2005)CrossRefGoogle Scholar
  45. 45.
    Bhangale, R.K., Ganesan, N.: Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates. Int. J. Solids Struct. 43(10), 3230–3253 (2006)zbMATHCrossRefGoogle Scholar
  46. 46.
    Guo, J.H., Chen, J.Y., Pan, E.: Size-dependent behavior of functionally graded anisotropic composite plates. Int. J. Eng. Sci. 106, 110–124 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Daneshmehr, A., Rajabpoor, A., Pourdavood, M.: Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions. Int. J. Eng. Sci. 82(3), 84–100 (2014)zbMATHCrossRefGoogle Scholar
  48. 48.
    Khorshidin, K., Fallah, A.: Buckling analysis of functionally graded rectangular nano-plate based on nonlocal exponential shear deformation theory. Int. J. Mech. Sci. 113(2), 94–104 (2016)CrossRefGoogle Scholar
  49. 49.
    Jandaghian, A.A., Rahmani, O.: Vibration analysis of functionally graded piezoelectric nanoscale plates by nonlocal elasticity theory: an analytical solution. Superlattices Microstruct. 100, 57–75 (2016)CrossRefGoogle Scholar
  50. 50.
    Pan, E.: Exact solution for functionally graded anisotropic elastic composite laminates. J. Compos. Mater. 37(21), 1903–1920 (2003)CrossRefGoogle Scholar
  51. 51.
    Lu, P., Zhang, P.Q., Lee, H.P., Wang, C.M., Reddy, J.N.: Non-local elastic plate theories. Philos. R. Soc. A Math. Phys. 463, 3225–3240 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Sarrami-Foroushani, S., Azhari, M.: Nonlocal vibration and buckling analysis of single and multi-layered graphene sheets using finite strip method including van der Waals effects. Physica E 57(3), 83–95 (2014)CrossRefGoogle Scholar
  53. 53.
    Pan, E.: Exact solution for simply supported and multilayered magneto-electro-elastic plates. J. Appl. Mech. 68(4), 608–618 (2001)zbMATHCrossRefGoogle Scholar
  54. 54.
    Zhong, Z., Shang, E.T.: Exact analysis of simply supported functionally graded piezo-thermoelectric plates. J. Intell. Mater. Syst. Struct. 16(16), 643–651 (2005)CrossRefGoogle Scholar
  55. 55.
    Wang, X., Sudak, L.J., Pan, E.: Pattern instability of functionally graded and layered elastic films under van der Waals forces. Acta Mech. 198, 65–86 (2008)zbMATHCrossRefGoogle Scholar
  56. 56.
    Li, Y., Yang, L., Zhang, L., Gao, Y.: Size-dependent effect on functionally graded multilayered two-dimensional quasicrystal nanoplates under patch/uniform loading. Acta Mech. 229, 3501–3515 (2018). MathSciNetCrossRefGoogle Scholar
  57. 57.
    Fan, T.Y.: Mathematical theory and methods of mechanics of quasicrystalline materials. Engineering 5, 407–448 (2013)CrossRefGoogle Scholar
  58. 58.
    Sun, T.Y., Guo, J.H., Zhang, X.Y.: Static deformation of a multilayered one-dimensional hexagonal quasicrystal plate with piezoelectric effect. Appl. Math. Mech. 39(3), 335–352 (2018)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MechanicsInner Mongolia University of TechnologyHohhotChina

Personalised recommendations