International Conference on Applied Anaylsis and Mathematical Modeling, İstanbul, Turkey, 10 - 13 March 2019, vol.1, no.1, pp.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 [3]. 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
[4] 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
[4] 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 [4] 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.
References
[1] Cemal
Eringen, Erdogan S. Suhubi, “Elastodynamics.
Vol I. Finite Motion; Vol II. Linear Theory”, Academic Press, 1975.
[2] Alexsandr
N. Guz, “Elastic Waves in Bodies with Initial (Residual) Stresses”, Kiev;
A.C.K. (in Russian), 2004.
[3] 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.