Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

doi:10.1007/s10483-015-1980-9

Abstract

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.

Key words: magnetoelectric (ME) material, functionally graded material (FGM), analytical solution, cantilever beam, plane stress problem

2010 MSC Number:

Yanmei YUE, Xiaofen YE, Kaiyu XU. Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam. Applied Mathematics and Mechanics (English Edition), 2015, 36(7): 955-970.

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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)
[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)
[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)
[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)
[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)
[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)
[7] Spaldin, N. A. and Fiebig, M. The renaissance of magnetoelectric multiferroics. Science, 309, 391-392 (2005)
[8] Eerenstein, W., Mathur, N. D., and Scott, F. Multiferroic and magnetoelectric materials. nature, 442, 759-765 (2006)
[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)
[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)
[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)
[12] Ichikawa, K. Functionally Graded Materials in 21st Century: a Workshop on Trends and Forecasts, Springer, New York, 18-20 (2000)
[13] Hirai, T. and Chen, L. Recent and prospective development of functionally graded materials. Japanese Material Science Forum, 509, 308-311 (1999)
[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)
[15] Hart, N. T., Brandon, N. P., Day, M. J., and Lapeña-Rey, N. Functionally graded composite cathodes for solid oxide fuel cells. Journal of Power Sources, 106, 42-50 (2002)
[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)
[17] Petrov, V. M. and Srinivasan, G. Enhancement of magnetoelectric coupling in functionally graded ferroelectric and ferromagnetic bilayers. Physical Review B, 78, 184421 (2008)
[18] Wang, R. F. and Pan, E. Three-dimensional modeling of functionally graded multiferroic composites. Mechanies of Advanced Material Structures, 18, 68-76 (2011)
[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)
[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)
[21] Shi, Z. F. General solution of a density functionally gradient piezoelectric cantilever and its applications. Smart Material and Structures, 11, 122-129 (2002)
[22] Shi, Z. F. and Chen, Y. Functionally graded piezoelectric cantilever beam under load. Archive of Applied Mechanics, 74, 237-247 (2004)
[23] Zhong, Z. and Yu, T. Analytical solution of a cantilever functionally graded beam. Composites Science and Technology, 67, 481-488 (2007)
[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)
[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)
[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)
[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)
[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)