International Conference on Applied Anaylsis and Mathematical Modeling, İstanbul, Türkiye, 10 - 13 Mart 2019, cilt.1, no.1, ss.44
In this work, a free vibration of a pre-stressed multilayered hollow sphere filled with a fluid is studied. The materials of each-layer of the sphere are assumed to be homogeneous and isotropic and, fluid is assumed to be compressible barotropic inviscid one. In addition, inhomogeneous initial stress in the multilayered sphere is assumed to be caused by the inner and/or outer uniform compressive radial forces. The investigations are made within the scope of the three-dimensional linearized theory of elastic waves in initially stressed bodies [1, 2]. However, the motion of the fluid is written within the scope of the linearized Navier-Stokes equations for compressible barotropic inviscid fluids . It is assumed that the ideal contact conditions on the interface surfaces between the neighboring layers and compatibility conditions on the contact surface between the sphere and fluid are satisfied. The predecessor of the present work is the investigation carried out in the paper  in which it was studies the natural vibration of the hollow sphere with inhomogeneous initial stresses caused by the internal and external hydrostatic pressure. Consequently, in the present work the problem considered in the paper  develops in two aspects: the first of which is the layering of the sphere and the second aspect is the fluid containing, and the solution procedure of this problem has two stage. In the first stage it is solved a static problem linear theory of elasticity on the stress distribution in the multilayered sphere under action of the hydrostatic pressure on the inner and outer surfaces of that. However, in the second stage it is solved the problem of the three-dimensioned natural vibration problem for this sphere within the scope of the 3D-linearized theory of elastic waves in initially stressed bodies. The equations used in the second stage contain the expressions of the initial stresses determined in the first stage and for solving these equation the discrete-analytical method developed in the paper in  is employed and it succeeds to obtain the analytical expressions for the amplitudes of the sought values through the spherical Bessel functions.
The solution to the corresponding equations for fluid motion is also found analytically in the spherical coordinates through the spherical Bessel functions. Using the traction free condition outer surface of the hollow sphere, the contact conditions between the layers of the sphere and compatibility conditions between the fluid and sphere on the inner surface of that the frequency equation is obtained for determination of the natural frequencies of the hydro-elastic system under consideration.
The aforementioned frequency equation is solved numerically and the numerical results illustrated the influence of the existence of the initial stresses as well as the existence of the fluid on the natural frequencies of the hydro-elastic system are presented and discussed.
 Cemal Eringen, Erdogan S. Suhubi, “Elastodynamics. Vol I. Finite Motion; Vol II. Linear Theory”, Academic Press, 1975.
 Alexsandr N. Guz, “Elastic Waves in Bodies with Initial (Residual) Stresses”, Kiev; A.C.K. (in Russian), 2004.
 Alexsandr N. Guz, “Dynamics of compressible viscous fluid”, Cambridge Scientific Publishers, 2009.
Surkay D. Akbarov, Hatam H. Guliyev, Yusif M.
Sevdimaliyev and Nazmiye Yahnioglu, “The Discrete-Analytical Solution Method
for Investigation Dynamics of the Sphere with Inhomogeneous Initial Stresses”, CMC,
55(2), pp.359-380, 2018.