Two-dimensional magnetoelastic problems of a thin current-carrying plate under the interaction of an unsteady electromagnetic field and a mechanical field are studied. The nonlinear magnetoelastic kinetic equations, geometric equations, physical equations, electrodynamics equations, and expressions for the Lorentz force of a thin current-carrying plate under the action of a coupled field are given. The normal Cauchy form nonlinear differential equations, which include ten basic unknown functions in all, are obtained by the variable replacement method. Using the difference and quasi-linearization methods, the nonlinear magnetoelastic equations are reduced to a sequence of quasilinear differential equations, which can be solved by the discrete-orthogonalization method. Numerical solutions for the magnetoelastic stresses and strains in a thin current-carrying elastic plate are obtained by considering a specific example. The results that the stresses and strains in the thin current-carrying elastic plate change with variation of the electromagnetic parameters are discussed. The results show that the stress–strain state in thin plates can be controlled by changing the electromagnetic parameters. This provides a method of theoretical analysis and numerical calculation for changing the service conditions and intensity research of thin plates of engineering structures in the electromagnetic field
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Knopoff, “The interaction between elastic wave motions and a magnetic field in electrical conductors,” J. Geoph. Research, 60, No. 4, 441–456 (1955).
P. Chadwick, “Elastic wave in thermoelasticity and magnetothermoelasticity,” Int. J. Eng. Sci., 10, No. 5, 143–153 (1972).
Y. H. Pao and C. S. Yeh, “A linear theory for soft ferromagnetic elastic bodies,” Int. J. Eng. Sci., 11, No. 4, 415–436 (1973).
S. A. Ambartsumyan, G. E. Bagdasaryan, and M. V. Belubekyan, Magnetoelasticity of Thin Shells and Plates [in Russian], Nauka, Moscow (1977).
F. C. Moon, Magneto-Solid Mechanics, John Wiley & Sons, New York (1984).
A. A. F. Van de Ven and M. J. H. Couwenberg, “Magneto-elastic stability of a superconducting ring in its own field,” J. Eng. Math., 20, 251–270 (1986).
A. T. Ulitko, L. V. Mol’chenko, and V. F. Kovalchuk, Magnetoelasticity under Dynamic Loading: A Workbook [in Ukrainian], Lybid’, Kyiv (1994).
L. V. Mol’chenko and Ya. M. Grigorenko, Fundamental Theory of Magnetoelasticity for Elements of Thin Plates and Shells: A Textbook [in Ukrainian], Kyiv University (2010).
Y. H. Zhou, Y. W. Gao, and X. J. Zheng, “Buckling and post-buckling of a ferromagnetic beam-plate induced by magneto-elastic interactions,” Int. J. Non-Lin. Mech., 35, No. 6, 1059–1065 (2000).
D. J. Hasanyan, G. M. Khachaturyan, and G. T. Piliposyan, “Mathematical modeling and investigation of nonlinear vibration of perfectly conductive plates in an inclined magnetic field,” Thin-Walled Structures, 39, No. 1, 111–123 (2001).
D. J. Hasanyan, L. Librescu, and D. R. Ambur, “Buckling and postbuckling of magnetoelastic flat plates carrying an electric current,” Int. J. Solids Struct., 43, No. 16, 4971–4996 (2006).
X. Z. Wang, J. S. Lee, and X. J. Zheng, “Magneto-thermo-elastic instability of ferromagnetic plates in thermal and magnetic fields,” Int. J. Solids Struct., 40, No. 22, 6125–6142 (2003).
X. Z. Wang, “Changes in the natural frequency of a ferromagnetic rod in a magnetic field due to magnetoelastic interaction,” Appl. Math. Mech., 29, No. 8, 1023–1032 (2008).
X. J. Zheng, J. P. Zhang, and Y. H. Zhou, “Dynamic stability of a cantilever conductive plate in transverse impulsive magnetic field,” Int. J. Solids Struct., 42, No. 8, 2417–2430 (2005).
G. Y. Wu, “The analysis of dynamic instability on the large amplitude vibrations of a beam with transverse magnetic fields and thermal loads,” J. Sound Vibr., 302, No. 1/2, 167–177 (2007).
M. Ottenio, M. Destrade, and R. W. Ogden, “Incremental magnetoelastic deformations with applications to surface instability,” J. Elasticity, 90, 19–42 (2008).
S. A. Kaloerov and O. I. Boronenko, “Two-dimensional magnetoelastic problem for a multiply connected piezomagnetic body,” Int. Appl. Mech., 41, No. 10, 1138–1148 (2005).
Yu. N. Podil’chuk and O. G. Dashko, “Stress state of a soft ferromagnetic with an ellipsoidal cavity in a homogeneous magnetic field,” Int. Appl. Mech., 41, No. 3, 283–290 (2005).
L. V. Mol’chenko, “Nonlinear deformation of current-carrying plates in a non-steady magnetic field,” Int. Appl. Mech., 26, No. 6, 555–558 (1990).
L. V. Mol’chenko, “Nonlinear deformation of shells of rotation of an arbitrary meridian in a non-stationary magnetic field,” Int. Appl. Mech., 32, No. 3, 173–179 (1996).
L. V. Mol’chenko and I. I. Loos, “Nonlinear deformation of conical shells in magnetic fields,” Int. Appl. Mech., 33, No. 3, 221–226 (1997).
L. V. Mol’chenko, “Influence of an external electric current on the stress state of an annular plate of variable stiffness,” Int. Appl. Mech., 37, No. 12, 108–112 (2001).
L. V. Mol’chenko, “A method for solving two-dimensional nonlinear boundary-value problems of magnetoelasticity for thin shells,” Int. Appl. Mech., 41, No. 5, 490–495 (2005).
L. V. Mol’chenko, I. I. Loos, and R. Sh. Indiaminov, “Determining the stress state of flexible orthotropic shells of revolution in magnetic field,” Int. Appl. Mech., 44, No. 8, 882–891 (2008).
L. V. Mol’chenko, I. I. Loos, and R. Sh. Indiaminov, “Stress–strain state of flexible ring plates of variable stiffness in a magnetic field,” Int. Appl. Mech., 45, No. 11, 1236–1242 (2009).
L. V. Mol’chenko, I. I. Loos, and I. V. Plyas, “Stress analysis of a flexible ring plate with circumferentially varying stiffness in a magnetic field,” Int. Appl. Mech., 46, No. 5, 567–572 (2010).
L. V. Mol’chenko, Nonlinear Magnetoelasticity of Thin Current-Carrying Shells [in Russian], Vyshcha Shkola, Kyiv (1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Prikladnaya Mekhanika, Vol. 51, No. 1, pp. 130–144, January–February 2015.
Rights and permissions
About this article
Cite this article
Bian, YH. Analysis of Nonlinear Stresses and Strains in a Thin Current-Carrying Elastic Plate. Int Appl Mech 51, 108–120 (2015). https://doi.org/10.1007/s10778-015-0677-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-015-0677-7