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# Effect of rotation, magnetic field and a periodic loading on radial vibrations thermo-viscoelastic non-homogeneous media

- Khalil S Al-Basyouni
^{1}Email author, - Samy R Mahmoud
^{1, 2}and - Ebraheem O Alzahrani
^{1}

**2014**:166

https://doi.org/10.1186/s13661-014-0166-7

© Al-Basyouni et al.; licensee Springer 2014

**Received:**16 April 2014**Accepted:**25 June 2014**Published:**24 September 2014

## Abstract

In this paper, an analytical solution for the effect of the rotation in a magneto-thermo-viscoelastic non-homogeneous medium with a spherical cavity subjected to periodic loading is presented. The distribution of displacements, temperature, and stresses in the non-homogeneous medium in the context of generalized thermo-elasticity using GL (Green-Lindsay) theory is discussed and obtained in analytical form. The results are displayed graphically to illustrate the effect of rotation, relaxation, magnetic field, viscoelasticity, and non-homogeneity. Comparisons are made with the previous work in the absence of rotation and initial stress.

## Keywords

- rotation
- viscoelasticity
- periodic loading
- non-homogeneity
- magneto-thermo-elasticity

## 1 Introduction

In recent years, the theory of magneto-thermo-viscoelasticity, which deals the interactions among strain, temperature, and electromagnetic fields has drawn the attention of many researchers because of its extensive use in diverse fields, such as geophysics for understanding the effect of the Earth’s magnetic field on seismic waves, damping of acoustic waves in a magnetic field, emission of electromagnetic radiations from nuclear devices, development of a highly sensitive superconducting magnetometer, electrical power engineering, optics, *etc.*; see [1]–[4]. Mahmoud *et al*. [5], [6] investigated the effect of the rotation on plane vibrations in a transversely isotropic infinite hollow cylinder, the effect of the rotation on wave motion through a cylindrical bore in a micropolar porous cubic crystal and he investigated the effect of a magnetic field and non-homogeneity on the radial vibrations in a hollow rotating elastic cylinder. Abd-Alla *et al*. [7]–[10] investigated the effect of the rotation on a non-homogeneous infinite cylinder of orthotropic material, influence of rotation, radial vibrations in a non-homogeneous orthotropic elastic hollow sphere subjected to rotation, and they investigated the magneto-thermo-elastic problem in a rotating non-homogeneous orthotropic hollow cylinder in the hyperbolic heat conduction model. Mahmoud [11], [12] investigated the analytical solution for an electrostatic potential on wave propagation modeling in human long wet bones, and they studied the influence of rotation and generalized magneto-thermo-elastics on Rayleigh waves in a granular medium under the effect of initial stress and a gravity field. Abd-Alla and Mahmoud [13] investigated the analytical solution of wave propagation in non-homogeneous orthotropic rotating elastic media. Abd-Alla *et al*. [14]–[16] investigated some problems like the propagation of an S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under the influence of a gravity field, the generalized magneto-thermo-elastic Rayleigh waves in a granular medium under the influence of a gravity field and initial stress, and they also investigated the problem of transient coupled thermo-elasticity of an annular fin. Some problems of thermo-elasticity and wave propagation modeling in a cylinder are investigated by Abd-Alla *et al*. [17], [18], respectively. Mukhopadhyay [19] investigated the effects of thermal relaxations on thermo-viscoelastic interactions in an unbounded body with a spherical cavity subjected to a periodic loading on the boundary. The effects of thermal relaxations on thermo-elastic interactions in an unbounded body with a spherical cavity or cylindrical hole subjected to a periodic loading on the boundary, respectively, were investigated by Roychoudhuri and Mukhopadhyay [20]. The thermally induced vibrations in a generalized thermo-elastic solid with a cavity have been investigated by Erbay *et al*. [21] and Li and Qi [22]. Mahmoud [23] investigated the analytical solution for free vibrations of an elasto-dynamic orthotropic hollow sphere under the influence of rotation.

In this paper, rotation and the magneto-thermo-elastic equation of a spherical cavity are decomposed into a non-homogeneous equation with boundary conditions. The effect of thermal relaxation times on the wave propagation in the magneto-thermo-viscoelastic case using the GL theory will be discussed. We take the material of the spherical cavity to be of Kelvin-Voigt type. Thus, the exact expressions for the transient response of displacement, stresses, and temperature in a spherical cavity are obtained. The numerical calculations will be investigated for the displacement, temperature, and the components of stresses, and we explain the special case from this study when the magnetic field and non-homogeneity are neglected. Finally, numerical results are calculated and discussed.

## 2 Formulation of the problem

where $\overline{h}$ is the perturbed magnetic field over the primary magnetic field, $\overline{E}$ is the electric intensity, $\overline{J}$ is the electric current density, ${\mu}_{e}$ is the magnetic permeability, $\overline{H}$ is the constant primary magnetic field, and $\overline{u}$ is the displacement vector.

Let ${c}_{0}=\frac{{\lambda}_{0}}{({\lambda}_{0}+2{\mu}_{0})}$, ${c}_{2}=\frac{{\gamma}_{0}}{({\lambda}_{0}+2{\mu}_{0})}$, ${c}_{3}=\frac{{\mu}_{e}^{0}{H}_{\varphi}^{2}}{({\lambda}_{0}+2{\mu}_{0})}$, ${c}_{v}=\sqrt{\frac{({\lambda}_{0}+2{\mu}_{0})}{{\rho}_{0}}}$.

In the following discussion the primes are neglected for ${r}^{\prime}$.

where ${c}_{4}=\frac{{\lambda}_{0}+{P}_{0}^{\ast}}{({\lambda}_{0}+2{\mu}_{0})}$, ${c}_{5}=\frac{{\gamma}_{0}}{({\lambda}_{0}+2{\mu}_{0})}$.

where ${l}_{1}=\frac{a{c}_{v}}{l}$, ${l}_{2}=\frac{a{\gamma}_{0}}{{\rho}_{0}}$.

## 3 The problem solution

where ${\mathrm{\nabla}}^{2}=\frac{{d}^{2}}{d{r}^{2}}+\frac{2}{r}\frac{d}{dr}$, ${\beta}_{1}={l}_{1}({\omega}^{2}{\tau}_{1}^{\prime}-i\omega )$, ${\beta}_{2}=i\omega {l}_{2}$.

where ${\mathrm{\Gamma}}_{1}={m}_{1}^{2}+{\beta}_{1}-\epsilon {\beta}_{2}$, ${\mathrm{\Gamma}}_{2}={m}_{1}^{2}+{\beta}_{1}{\eta}_{1}-\epsilon {\beta}_{2}$.

## 4 Boundary conditions

This is the solution of the current problem for the case of a non-homogeneous isotropic viscoelastic unbounded body with spherical cavity without the effect of a magnetic field; it coincides with one previously published.

## 5 Discussion and numerical results

## 6 Conclusions

The elasto-dynamic equations for the generalized thermo-viscoelastic theory under the effect of the non-homogeneous material, rotation, relaxation, and magnetic field have a complicated nature. The method used in this study provides a quite successful approach in dealing with such problems. The displacement, temperature, and stress components have been obtained in analytical form. This approach gives an exact solution in the Hankel transform domain that appears in the governing equations of the problem considered. Numerical results are calculated and discussed and illustrated graphically.

## Declarations

### Acknowledgements

This article (project) was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-094-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.

## Authors’ Affiliations

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