Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam. (English) Zbl 1322.74039

Summary: 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.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74E30 Composite and mixture properties
74F15 Electromagnetic effects in solid mechanics
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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[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) · Zbl 1158.74392
[2] Luo, X. B.; Wu, D.; Zhang, N., Room temperature magneto-birefringence in composites of stress-birefringence and magnetostriction, Journal of Applied Physics, 113, 173903, (2013)
[3] Zhong, X. C.; Lee, K. Y., Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties, Archive of Applied Mechanics, 82, 791-807, (2012) · Zbl 1293.74390
[4] Hadjiloizi, D. A.; Georgiades, A. V.; Kalamkarov, A. L.; Jothi, S., Micromechanical modeling of piezo-magneto-thermo-elastic composite structures: part I—theory, European Journal of Mechanics-A/Solids, 39, 298-312, (2013) · Zbl 1348.74268
[5] Fang, F.; Shan, S. C.; 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) · Zbl 06197638
[6] Dong, S. X.; Li, J. F.; Viehland, D., Vortex magnetic field sensor based on ring-type magnetoelectric laminate, Applied Physics Letters, 85, 2307-2309, (2004)
[7] Spaldin, N. A.; Fiebig, M., The renaissance of magnetoelectric multiferroics, Science, 309, 391-392, (2005)
[8] Eerenstein, W.; Mathur, N. D.; Scott, F., Multiferroic and magnetoelectric materials, nature, 442, 759-765, (2006)
[9] Srinivasan, G.; Zavislyak, I. V.; 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.; Feng, W. J., Effective properties of multilayered functionally graded multiferroic composites, Composite Structures, 87, 206-214, (2009)
[11] Sladek, J.; Sladek, V.; Krahulec, S.; 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.; 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.; 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) · Zbl 1110.74748
[15] Hart, N. T.; Brandon, N. P.; Day, M. J.; Lape˜na-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.; Hempel, D., Functionally graded materials for biomedical applications, Material Science and Engineering A, 362, 40-60, (2003)
[17] Petrov, V. M.; Srinivasan, G., Enhancement of magnetoelectric coupling in functionally graded ferroelectric and ferromagnetic bilayers, Physical Review B, 78, 184421, (2008)
[18] Wang, R. F.; 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.; 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) · Zbl 0954.74077
[20] Ashrafi, H.; Asemi, K.; 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) · Zbl 1406.74075
[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.; Chen, Y., Functionally graded piezoelectric cantilever beam under load, Archive of Applied Mechanics, 74, 237-247, (2004) · Zbl 1119.74466
[23] Zhong, Z.; Yu, T., Analytical solution of a cantilever functionally graded beam, Composites Science and Technology, 67, 481-488, (2007)
[24] Pan, E.; 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.; Chen, W. Q., Elasticity solutions for plane anisotropic functionally graded beams, International Journal of Solids and Structures, 44, 176-196, (2007) · Zbl 1155.74369
[26] Huang, D. J.; Ding, H. J.; 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.; 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.; 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)
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