^{1}

^{2}

^{*}

^{3}

^{4}

Torsional vibrations of coated hollow poroelastic spheres are studied employing Biot’s theory of wave propagation in poroelastic solid. The dilatations of solid and liquid media are zero, therefore the frequency equation of torsional vibrations is same both for a permeable and an impermeable surface. The coated poroelastic sphere consists of an inner hollow poroelastic sphere bounded by and bonded to a sphere made of distinct poroelastic material. The inner sphere is designated as core and outer sphere as casing. Core and casing are bonded at the curved surfaces. The inner and outer boundaries of the coated hollow poroelastic sphere are free from stress and at the interface of core and casing the displacement and stresses are continuous. It is assumed that the each material of coated sphere is homogeneous and isotropic. The frequency equation of torsional vibrations of a coated poroelastic hollow sphere is obtained when the material of the core vanishes. Also a coated poroelastic solid sphere is obtained as the limiting case of the frequency equation of coated hollow poroelastic sphere when the inner radius of core approaches to zero. Non-dimensional frequency as a function of ratio of thickness of core to that of inner radius of core is determined and analyzed. It is observed that the frequency and dispersion increase with the increase of the thickness of the coating.

Wave propagation is the phenomenon of energy transfer. Due to stress wave propagation the cracks are developed at the surface. To avoid the development of cracks on the surface of the material, the coating is provided on the material. Coating material is chosen with good tribological properties. The nature of contact between different components determines the state of stress which controls the fretting. The coating delays the crack initiation and retards the crack propagation. Paul [

In the present analysis, torsional vibrations of coated hollow poroelastic spheres are studied. The dilatations of solid and liquid media are zero, hence the liquid pressure developed in solid-liquid aggregate is zero so that the frequency equation of torsional vibrations is same both for a permeable and an impermeable surface. The frequency equation of a coated poroelastic solid sphere is obtained as a limiting case of coated poroelastic hollow sphere. The plots of frequency as a function of ratio of thickness of core to inner radius are presented for two types of coated hollow poroelastic spheres. There is increase in dispersion with the increase in thickness of the coating for the considered coated sphere. The torsional waves are non dispersive in thin coated poroelastic hollow sphere. The results of purely elastic solid are shown as a special case.

The study of torsional vibrations of elastic solid is important in several fields, e.g., soil mechanics, transmission of power through shafts with flange at the end as integral part of the shaft. It is now recognized that virtually no high-speed equipment can be properly designed without obtaining solution to what are essentially lateral or torsional vibration problems. Examples of torsional vibrations are vibrations in gear train and motor-pump shafts. Thus, from engineering point of view the study of torsional vibrations has greater interest. Such vibrations, for example, are used in delay lines. Further, based on reflections and refractions during the propagation of a pulse imperfection can be identified. This investigation is particularly applicable in Bio-Mechanics to identify and study the cracks in bones.

The equations of motion of a poroelastic solid [

where

where

Let (_{1} and the outer radius of casing is r_{2} and also the interface lie at^{*}. The outer and inner surfaces of the coated poroelastic sphere are free from stress and at the interface, the stresses and displacements are continuous. For torsional vibrations, the only non-zero displacement components of the solid and liquid media are

where

Let the propagation mode shapes of solid and liquid

here

where

The second equation of (5) gives

Using Equation (7) into first equation of (5), we obtain

where

In Equation (9),

where

Thus the displacement of solid is

From Equation (4), it can be seen that the normal strains

The stresses and displacements of solid for the outer spherical shell (coating) and inner spherical shell (core) are

where the elements

Outer and inner surfaces of the coated hollow poroelastic sphere are assumed to free from stress and at the interface, the stresses and displacements are continuous. Thus the boundary conditions for the considered problem are

Since the considered vibrations are shear vibrations, the dilatations of solid and liquid media each is zero, thereby liquid pressures in outer and inner poroelastic spherical shells _{1}, C_{2}, D_{1}, D_{2} we get the frequency equation

In Equation (16), the elements

Now we consider two particular cases of the frequency Equation (16) in the following:

(i) When inner radius of core approaches to zero, the considered problem reduces to the problem of torsional wave propagation in solid coated poroelastic sphere. In this case, the frequency Equation (16) under suitable boundary conditions reduce to

where the elements

(ii) When the poroelastic material of the core vanishes, the considered problem reduces to the problem of torsional wave propagation in hollow poroelastic sphere and the frequency Equation (16) or (17) under suitable boundary conditions, reduce to

By using the following relations [

Equation (18) is simplified to the form [Equation (17), Bulletin of Calcutta Mathematical Society, Volume.103, pp.161-170] presented in and studied by Ahmed Shah and Tajuddin [

For the purpose of analysis of the frequency Equation (16), we consider a non- dissipative medium where b = 0. It is convenient to introduce the following non- dimensional variables:

where

Frequency equation of coated poroelastic spheres (16) is non-dimensionalised. For a given poroelastic material, Equation (16) constitutes a relation between non-dimensional frequency and ratio of thickness of core to inner radius for fixed values of_{2}, viz., 1.1, 1.5 and 3.0 are taken for numerical computation. These values of

The phase frequency first three modes of coated poroelastic spheres-I and II are presented in

sphere-I and II are dispersive. Hence it can be concluded that modes are non dispersive in thin coated spheres and as the thickness increases, the modes become dispersive. We also see there is increase in the frequency with the increase in thickness of the coating.

Shah, S.A., Nageswaranath, C., Ramesh, M. and Ramanamurthy, M.V. (2017) Torsional Vibrations of Coated Hollow Poroelastic Spheres. Open Journal of Acoustics, 7, 18-26. https://doi.org/10.4236/oja.2017.71003