Investigation of flexural vibrations in poroelastic solid sphere in the presence of static stresses

In this paper, the three dimensional wave propagation in a homogeneous isotropic poroelastic solid sphere is investigated in the presence of static stresses. The frequency equation is obtained in the framework of Biot's theory. Non-dimensional phase velocity is computed as a function of uniaxial static stress for fixed wavenumber. For the numerical results, three types of poroelastic spherical solids are employed, and the results are presented graphically.


Introduction
The analysis for transient problems of spherical structures is important and interesting research fields for engineers and scientists. The applications for homogeneous solid sphere have continuously increased in some engineering areas, including aerospace, frequency control filters, chemical vessels, information storage devices, and signal processing devices. Even in the human body most of the bones are in spherical shape. Mott investigated elastic motion of an isotropic medium in the presence of body forces and static stresses [1]. In the paper [1], Mott derived equations of elastic motion which include the effect of static surface forces and body forces. When the body forces imposed on a solid which is at rest gives static stresses and strains. Effect of static axial stress upon the velocity of the lowest-order flexural mode is investigated in [2]. Employing Biot's theory of poroelasticity [3], detailed investigation of wave propagation is given in [4]. From the mathematics perspective of poroelastic materials, several authors are investigated wave propagation phenomena which characterize Biot's theory [5][6][7][8][9]. Flexural vibrations in poroelastic elliptic cone against the angle made by the major axis of the cone in spheroconal coordinate system are given in [10]. Shah and Tajuddin [11] discussed torsional vibrations of poroelastic spheroidal shells. In this paper, authors derived frequency equations for poroelastic thin spherical shell, thick spherical shell, poroelastic solid sphere, and concluded that the frequency is same for all the three cases. Axially symmetric vibrations of fluid filled poroelastic spherical shell are studied by Shah and Tajuddin [12]. In the said paper, frequency equations are derived for radial and rotatory vibrations of fluid-filled and empty poroelastic spherical shells. A comparative study is made between the modes of composite spherical shell and its ring modes [13]. Shanker et. al. [14] investigated vibration analysis of a poroelastic composite hollow sphere. In this paper, authors derived frequency equations for radial and rotatory vibrations of fluid filled and empty poroelastic shells with rigid core. Torsional vibrations of thick-walled hollow poroelastic spheres are studied by Ahmed Shah and Tajuddin [15]. In this paper, authors derived complex frequencies of torsional vibrations of thick-walled hollow poroelastic spheres for different dissipations and it is concluded that as the dissipation increases, the propagation increases while the attenuation remains almost same. Flexural vibrations of poroelastic solids in presence of static stress are studied by Rajitha et al. [16]. In the paper, governing equations are derived in the presence of static stress, which were not available in the earlier literature. However, in all the above papers, static stress is not employed in the case of flexural vibrations of poroelastic solid sphere. Therefore, in this paper, the same is investigated in the framework of Biot's theory.
This paper is organized as follows. In section 2, governing equations, and solution of the problem are given. In section 3, the case of static uniaxial stress is considered. Numerical results are described in section 4. Finally, conclusion is given in section 5.

Governing equations and solution of the problem
Consider an isotropic poroelastic solid sphere in spherical coordinate system (r, θ, φ). Let − → u (u, v, w) and − → U (U, V, W ) be the solid and fluid displacements. When the body forces are large, displacements will be large. Consequently second order coupling between stresses and strains cannot be ignored. Effective stresses must be inserted in the place of usual ones. The effective stress-strain relations are as follows [1]: The expressions of effective stresses involve only shear components. As there are no shear components for the case of fluid pressure, the expression for the fluid pressure remains the same. In equation (1), usual stresses σ ij and fluid pressure s are given under [3].
In equation (2), e ij 's strain displacements, A, N, Q, R are poroelastic constants, e and ε are the dilatations of solid and fluid, respectively, and δ ij is the well-known Kronecker delta function. Substitution of effective stresses for usual stresses, the equations of motion obtained in the following manner.
(3) In equation (3), ρ ij are mass coefficients, F r , F θ , and F φ are the components of the body force vector − → F . Equations of motion and the boundary conditions together form the basis for the solution of wave propagation problems in poroelastic solids. The equations (2) and (3) have to be satisfied at every point of the poroelastic solid, and on surface of that solid. Thus the solution of the problem can be completely determined. To originate connection between the time variant and the static poroelastic quantities, the displacements and the stresses are written in the following form [1,16]: where ω n is the n th angular frequency. Using equations(1), (2) and (4) in the equation (3), we get the following static equations, (n = 0): Similarly, the harmonic equations n = 1 are obtained, which are: In all the above, F r 1 , F θ 1 , F φ 1 are the body forces which have the frequency ω. The equations (5) and (6) are static equations, and first harmonic equations respectively.

The case of uniaxial static stress
When the large static uniaxial stress is applied on a solid medium one can ignore the effect of body forces [1]. If we assume that applied static uniaxial stress is acting in the direction of φ− axis, then we have σ ij 0 = e ij 0 = 0, (i = j), σ rr 0 = σ θθ 0 = 0, where σ φφ 0 is the applied uniform static uniaxial stress. Substituting the equation (7) in the equation (5), it can be seen that equation (5) are automatically satisfied and the static strains are obtained as follows: e rr 0 = e θθ 0 = −ϑe φφ 0 .
In all the above C 1 , C 2 , C 3 , C 4 , C 5 , C 6 are arbitrary constants, j is the complex unity, and k i (i = 1, 2, 3) is the wave number in the i th direction such that the wavenumber k = k 2 1 + k 2 2 + k 2

Conclusion
Employing Biot's theory, flexural vibrations of poroelastic solid sphere in presence of static stress are investigated. Non-dimensional phase velocity against the non-dimensional static uniaxial stress for fixed wavenumber is computed for three types of poroelastic solids. Material-1 value is much greater than that of material-2 for the same value of applied uniaxial static stress. Both the materials are sandstone related and differ in only the fluid part. Hence, it can be inferred that the fluid part is causing the above discrepancy. This kind of analysis can be made for any poroelastic solid sphere if the values of parameters are available.