 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| A. G. Bagdoev and S. G. Saakyan, "Stability of nonlinear modulation waves in a magnetic field for the spatial and the averaged problems," Mech. Solids. 36 (5), 25-30 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
5 |
Pages |
25-30 |
| Title |
Stability of nonlinear modulation waves in a magnetic field for the spatial and the averaged problems |
| Author(s) |
A. G. Bagdoev (Erevan)
S. G. Saakyan (Erevan) |
| Abstract |
The linear problem of vibration of a magnetoelastic plate
in a transverse magnetic field is considered in [1] in the framework of
the Kirchhoff theory. A solution is obtained in the case of small
electrical conductivity on the basis of the hypothesis of
magnetoelasticity and without that hypothesis. In [2], this problem is
solved under the assumption that the projection of the induced magnetic
field onto the normal to the (nonperturbed) plate linearly depends on its
normal component. By means of complicated calculations, a simple linear
dispersion relation is obtained. In the case of small or large electrical
conductivity, this relation transforms into known solutions. In [3], a
linear solution is constructed for a plate of finite thickness. Nonlinear
problems of stability of modulation waves in a longitudinal magnetic
fields are solved in [4, 5]. The problems mentioned above have been
studied on the basis of the classical theory of plates. In the present
paper, two-dimensional modulation waves in a plate with finite
conductivity in a magnetic field are studied by the method of
simultaneously solving the equations of elasticity and those of magnetic
induction. First, we consider the linear spatial problem of
magnetoelasticity for a plate in a transverse magnetic field and obtain
the corresponding dispersion relation. In the elastic case, this relation
yields a known solution. In the case of small thickness and Alfven
velocity, a simple dispersion relation is obtained. Next, we study the
linear and nonlinear problems averaged over the thickness of the plate.
The dispersion relation is also obtained for the spatial and the averaged
problems for a plate in a longitudinal magnetic field. Conditions of
stability of nonlinear waves are studied. |
| References |
| 1. | S. A. Ambartsumyan, G. E. Bagdasaryan, and M. V. Belubekyan,
Magnetoelasticity of Thin Shells and Plates [in Russian], Nauka, Moscow, 1977. |
| 2. | S. A. Ambartsumyan and G. E. Bagdasaryan,
Electroconductive Plates and Shells in a Magnetic Field [in Russian],
Nauka, Moscow, 1996. |
| 3. | S. A. Ambartsumyan and M. V. Belubekyan,
Vibrations and Stability of Current-carrying Elastic Plates [in Russian],
National Academy of Sciences of Armenia, Erevan, 1992. |
| 4. | A. G. Bagdoev and L. A. Movsesyan,
"Nonlinear vibrations of plates in a longitudinal magnetic field,"
Izv. AN ArmSSR, Mekhanika, Vol. 35, No. 1, pp. 16-22, 1982. |
| 5. | A. G. Bagdoev and L. A. Movsesyan,
"Modulations of thermal magnetoelastic waves in nonlinear plates,"
Izv. AN ArmSSR, Mekhanika, Vol. 52, No. 1, pp. 25-30, 1999. |
| 6. | W. Nowacki, Theory of Elasticity [Russian translation], Mir, Moscow, 1975. |
| 7. | H. Kauderer, Nonlinear Mechanics [Russian translation], Izd-vo Inostr. Lit-ry,
Moscow, 1961. |
| 8. | R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morries,
Solitons and Nonlinear Wave Equations [Russian translation], Mir, Moscow, 1988. |
| 9. | V. I. Karpman, Nonlinear Waves in Dispersive Media [in Russian],
Nauka, Moscow, 1973. |
|
| Received |
20 January 2000 |
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