Magneto-elastic radial vibrations of a transversely isotropic hollow cylinder

Abstract

The radial vibrations of a thick transversely isotropic elastic cylinder subjected to a uniform axial magnetic field are considered. The problem is described by the equations of elasticity taking into account the effect of the magnetic field and the electro-magnetic equations of Maxwell. This requires the solution of the equations of motion in cylindrical coordinates with the z-axis directed along the axis of the cylinder. The frequency equations have been derived in the form of a determinant involving Bessel functions. The roots of the frequency equation give the values of the characteristic circular frequency parameters of the first four modes for various geometries. These roots, which correspond to various modes, are numerically calculated and presented graphically. This study shows that waves in a solid body propagating under the influence of a superimposed magnetic field can differ significantly from those propagating in the absence of a magnetic field. Also, the circular frequency increases with decreasing wall thickness of the cylindrical shell for all modes and increase with the higher modes of motion. Finally, it is noticed that the effect of the magnetic field on the mode frequency becomes less significant for the higher modes.

A mathematical model for longitudinal wave propagation in a magnetoelastic hollow circular cylinder of anisotropic material under the influence of initial hydrostatic stress

“…where α 2 = μ o H o 2 4 π ρ and α is the so-called Alfven wave velocity (see [17, p.472]), and the electric field intensity is given as the form…”

“…The elements X ¯ ij are given by (32) with q → 0 . The equation Δ 1 = 0 represents a motion involving the radial displacement u only, corresponding to the radial vibrations [17]. Δ 2 = 0 represents a motion involving the axial displacement w only, corresponding to the axial shear vibrations [8,15].…”

“…Among many important problems that are considered in such studies, the problems of elastic wave propagation in the presence of a steady magnetic field have been investigated when the material is isotropic and homogeneous by Andreou and Dassios [11], Das and Bhattacharya [12], Gourakishwar [13], Paria [14] and Suhubi [15]. Some of the analogous results on magnetoelastic wave propagation problems, but in an anisotropic medium, were obtained by Abd-alla [16,17] and Datta [18]. General details and many references on these subjects may be found in monographs by Eringen and Suhubi [19], Eringen and Maugin [20], Auld [21], Moon [22] and Nowacki [23].…”

The influence of hydrostatic stress on the frequency equation of flexural waves in a magnetoelastic transversly isotropic circular cylinder

“…the determinant equation , breaks into the product of sub determinants , which is satisfied if either or equal to zero, where The frequency equation , corresponds to magneto‐elastic radial waves if we put the initial stress equals zero. While the frequency equation , corresponds to the magneto‐elastic torsional waves and , where given in Eq. when …”

“…Development of magnetoelasticity also induces us to study various problems of geophysics, seismology and related topics . For example, Suhubi , Datta , Abd‐alla , and Abd‐alla et al. have been discussed a similar problems but in more general way where the magnetic field is taken in their considerations.…”