Abstract
The problems of nonlinear deformation of a thin current-carrying shell under the coupled action of an unsteady electromagnetic field and a mechanical field are studied. The nonlinear magneto-elastic kinetic equations, the physical equations, the geometric equations, the electrodynamics equations, and the expressions of Lorentz force of a thin current-carrying shell under the action of a coupled field are given. 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 quasilinearization methods, the nonlinear magneto-elastic equations are reduced to a sequence of quasilinear differential equations, which can be solved by the method of discrete orthogonalization. Numerical solutions for the stresses and deformations in the thin current-carrying strip-shell with two simply supported edges are obtained by considering a specific example. The results that the stresses and deformations in a thin current-carrying strip-shell with two simply supported edges change with variation of the electromagnetic parameters are discussed, through a special case. It is shown that the deformations of the shell can be controlled by changing the electromagnetic parameters
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. C. Eringen, “Unified continuum theory of electrodynamics of liquid crystals,” Int. J. Eng. Sci., 25, 12–13 (1987).
A. C. Eringen, “Theory of electric-magnetic plates,” Int. J. Eng. Sci., 27, No. 4, 63–75 (1989).
A. C. Eringen, “A mixture theory of electromagnetism and superconductivity,” Int. J. Eng. Sci., 36, No. 5, 525–543 (1998).
L. V. Mol’chenko, Mathematical Principle of Nonlinear Magnetoelasticity of Thin Plate [in Russian], Inst. Mat., Kiev (1988).
Y. Liang, Y. P. Shen, and M. Zhao, “Magnetoelastic formulation of soft ferromagnetic elastic problems with collinear cracks: Energy density fracture,” Theor. Appl. Fract. Mech., 34, 49–60 (2000).
E. Gaganidze, P. Esquinazi, and M. Ziese, “Dynamical response of vibrating ferromagnets,” J. Magnetism and Magnetic Materials, 210, No. 3, 49–62 (2000).
Y. H. Zhou, Y. W. Gao, and X. J. Zheng, “Buckling and post-buckling analysis for magneto-elastic-plastic ferromagnetic beam-plates with unmovable simple supports,” Int. J. Solids Struct., 40, 2875–2887 (2003).
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).
Hasanyan, Davresh, Librescu, and Liviu, “Nonlinear vibration of finitely electro-conductive plate-strips in a magnetic field,” in: Proc. Structural Dynamics and Materials Conf., AIAA/ASME/ASCE/AHS/ASC Structures, 4 (2003), pp. 2827–2837.
H. H. Sherie and K. A. Hekmy, “A two-dimensional problem for a half-space in magneto thermoelasticity with thermal relaxation,” Int. J. Eng. Sci., 40, 587–604 (2002).
M. A. Ezzat and A. S. El-Karamany, “Magnetothermoelasticity with two relaxation times in conducting medium with variable electrical and thermal conductivity,” Appl. Math. Comp., 142, 449–467 (2003).
Chun-Bo Lin and Chau-Shioung Yeh, “The magnetoelastic problem of a crack in a soft ferromagnetic solid,” Int. J. Solids Struct., 39, 1–17 (2002).
Chun-Bo Lin and Hsien-Mou Lin, “The magnetoelastic problem of cracks in the bonded dissimilar materials,” Int. J. Solids Struct., 39, 2807–2826 (2002).
Chun-Bo Lin, “On the bounded elliptic elastic inclusion in plane magnetoelasticity,” Int. J. Solids Struct., 40, 1547–1565 (2003).
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, The Nonlinear Magnetoelasticity of Thin Current-Carrying Plate [in Russian], Vyshch. Shkola, Kiev (1989).
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).
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).
Author information
Authors and Affiliations
Additional information
Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 130–144, September 2007.
Rights and permissions
About this article
Cite this article
Bian, YH., Tian, ZG. & Bai, XZ. Nonlinear stress and deformation analysis of thin current-carrying strip-shells. Int Appl Mech 43, 1057–1068 (2007). https://doi.org/10.1007/s10778-007-0107-6
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10778-007-0107-6