ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

parameters of impact such as maximum transferred force, maximum acceleration, and
the impact duration.
3. THEORY
The inclined impact of fluid filled spherical shell with a torus shape object with the
following conditions is considered. A fluid filled spherical shell with mass msk and
velocity Vsk, and a torus shape object with mass me and velocity Ve illustrated in figure
1. The outer radius of shell is Rsk and its thickness is h. The shell material properties
are considered as homogeneous and isotropic with Esk as its Young Modulus, vsk as
Poisson's Ratio, and ρsk as the density. The shell is filled with viscous fluid with
density ρf and bulk Modulus B. The outer radius of fluid is Rf, which is the inner
radius of the shell. The torus shape radii are Re and R e. The torus shape properties
are also considered as homogeneous and isotropic with Young Modulus Ee, Poisson's
Ratio νe, and Density ρe, The impact angle is α.
3-1- The laws of Impact
1- The contact resulting from the impact of two solid objects which is according to the
Hertzian contact theory.
2- The contact resulting from the impact to a thin hollow spherical shell.
With the assumption of Hertzian contact model, the relationship for forcedisplacement
of the two objects is given by Hertz [7]:
K F ∆ =
3
2
H H x
4
K H = qk . R . E
3
1
* 2 *
( 1)
Here F is the applied force, XH the mutual displacement of the objects, and KH the
contact stiffness.
Fig. 1: The oblique impact of the spherical shell and torus object, with the impact angle a
In equation (1):
1/R * =1/2(2/Rsh+1/Re+1/R e), 1/E * =1/π[(1-νe 2 )/Ee+(1-νsh 2 )/Esh] (2)
The peak displacement for the thin hollow spherical shell due to load F, as a uniform
pressure on a small sphere of radius a, can be explained as:
F = K . ∆x
sh sh
and Ksh can be calculated from the following equation: [8]
(3)