Skip to main content
SpringerLink
Log in

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Liu, L. P. An energy formulation of continuum magneto-electro-elasticity with applications. Journal of the Mechanics and Physics of Solids, 28, 560–568 (2009)

    Article  Google Scholar 

  2. Luo, X. B., Wu, D., and Zhang, N. Room temperature magneto-birefringence in composites of stress-birefringence and magnetostriction. Journal of Applied Physics, 113, 173903 (2013)

    Article  Google Scholar 

  3. Zhong, X. C. and Lee, K. Y. Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties. Archive of Applied Mechanics, 82, 791–807 (2012)

    Article  MATH  Google Scholar 

  4. Hadjiloizi, D. A., Georgiades, A. V., Kalamkarov, A. L., and Jothi, S. Micromechanical modeling of piezo-magneto-thermo-elastic composite structures: part I—theory. European Journal of Mechanics-A/Solids, 39, 298–312 (2013)

    Article  MathSciNet  Google Scholar 

  5. Fang, F., Shan, S. C., and Yang, W. Magnetoelectric coupling of Terfenol-D/P (VDF-TrFe) /Terfenol-D laminates mediated by crystallite size of electroactive polymer. Acta Mechanics, 224, 1169–1174 (2013)

    Article  MATH  Google Scholar 

  6. Dong, S. X., Li, J. F., and Viehland, D. Vortex magnetic field sensor based on ring-type magnetoelectric laminate. Applied Physics Letters, 85, 2307–2309 (2004)

    Article  Google Scholar 

  7. Spaldin, N. A. and Fiebig, M. The renaissance of magnetoelectric multiferroics. Science, 309, 391–392 (2005)

    Article  Google Scholar 

  8. Eerenstein, W., Mathur, N. D., and Scott, F. Multiferroic and magnetoelectric materials. nature, 442, 759–765 (2006)

    Article  Google Scholar 

  9. Srinivasan, G., Zavislyak, I. V., and Tatarenko, A. S. Millimeter-wave magnetoelectric effects in bilayers of barium hexaferrite and lead zirconate titanate. Applied Physics Letters, 89, 152508 (2006)

    Article  Google Scholar 

  10. Wang, X., Pan, E., Albrecht, J. D., and Feng, W. J. Effective properties of multilayered functionally graded multiferroic composites. Composite Structures, 87, 206–214 (2009)

    Article  Google Scholar 

  11. Sladek, J., Sladek, V., Krahulec, S., and Pan, E. Enhancement of the magnetoelectric conefficient in functionally graded multiferroic composites. Journal of Intelligent Material Systems and Structures, 23, 1649–1658 (2012)

    Article  Google Scholar 

  12. Ichikawa, K. Functionally Graded Materials in 21st Century: a Workshop on Trends and Forecasts, Springer, New York, 18–20 (2000)

    Google Scholar 

  13. Hirai, T. and Chen, L. Recent and prospective development of functionally graded materials. Japanese Material Science Forum, 509, 308–311 (1999)

    Google Scholar 

  14. Wang, Y. S., Huang, G. Y., and Dross, D. On the mechanical modeling of functionally graded interfacial zone with a Griffith crack: anti-plane deformation. Journal of Applied Mechanics, 70, 676–680 (2003)

    Article  MATH  Google Scholar 

  15. Hart, N. T., Brandon, N. P., Day, M. J., and Lape˜na-Rey, N. Functionally graded composite cathodes for solid oxide fuel cells. Journal of Power Sources, 106, 42–50 (2002)

    Article  Google Scholar 

  16. Pompe, W., Worch, H., Epple, M., Friness, W., Gelinsky, M., Greil, P., and Hempel, D. Functionally graded materials for biomedical applications. Material Science and Engineering A, 362, 40–60 (2003)

    Article  Google Scholar 

  17. Petrov, V. M. and Srinivasan, G. Enhancement of magnetoelectric coupling in functionally graded ferroelectric and ferromagnetic bilayers. Physical Review B, 78, 184421 (2008)

    Article  Google Scholar 

  18. Wang, R. F. and Pan, E. Three-dimensional modeling of functionally graded multiferroic composites. Mechanies of Advanced Material Structures, 18, 68–76 (2011)

    Article  Google Scholar 

  19. Ding, H. J., Wang, G. Q., and Chen, W. Q. A boundary integral formulation and 2D fundamental solution for piezoelectric media. Computure Methods in Applied Mechanics and Engineering, 158, 65–80 (1998)

    Article  MATH  Google Scholar 

  20. Ashrafi, H., Asemi, K., and Shariyat, M. A three-dimensional boundary element stress and bending analysis of transversely/longitudinally graded plates with circular cutouts under biaxial loading. European Journal of Mechanics-A/Solids, 42, 344–357 (2013)

    Article  MathSciNet  Google Scholar 

  21. Shi, Z. F. General solution of a density functionally gradient piezoelectric cantilever and its applications. Smart Material and Structures, 11, 122–129 (2002)

    Article  Google Scholar 

  22. Shi, Z. F. and Chen, Y. Functionally graded piezoelectric cantilever beam under load. Archive of Applied Mechanics, 74, 237–247 (2004)

    Article  MATH  Google Scholar 

  23. Zhong, Z. and Yu, T. Analytical solution of a cantilever functionally graded beam. Composites Science and Technology, 67, 481–488 (2007)

    Article  Google Scholar 

  24. Pan, E. and Han, F. Exact solution for functionally graded and layered magneto-electro-elastic plates. International Journal of Engineering Science, 43, 321–339 (2005)

    Article  Google Scholar 

  25. Ding, H. J., Huang, D. J., and Chen, W. Q. Elasticity solutions for plane anisotropic functionally graded beams. International Journal of Solids and Structures, 44, 176–196 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Huang, D. J., Ding, H. J., and Chen, W. Q. Analytical solution for functionally graded magnetoelectro- elastic plane beams. International Journal of Engineering Science, 45, 467–485 (2007)

    Article  Google Scholar 

  27. Li, X. Y., Ding, H. J., and Chen, W. Q. Three-dimensional analytical solution for functionally graded magneto-electro-elastic circular plates subjected to uniform load. Composite Structures, 83, 381–390 (2008)

    Article  Google Scholar 

  28. Huang, D. J., Ding, H. J., and Chen, W. Q. Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading. European Journal of Mechanics- A/Solids, 29, 356–369 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Department of Mechanics, Shanghai University, Shanghai, 200072, China

    Yanmei Yue, Xiaofen Ye & Kaiyu Xu

Authors
  1. Yanmei Yue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofen Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kaiyu Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kaiyu Xu.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 10772106 and 11072138), the Shanghai Leading Academic Discipline Project (No. S30106), the Research Fund for the Doctoral Program of Higher Education of China (No. 20113108110005), and the Natural Science Foundation Project of Shanghai (No. 15ZR1416100)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yue, Y., Ye, X. & Xu, K. Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam. Appl. Math. Mech.-Engl. Ed. 36, 955–970 (2015). https://doi.org/10.1007/s10483-015-1980-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-015-1980-9

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation