In this paper, a closed form solution of an arbitrary oriented hollow elastic ellipsoidal shell impacting with an elastic flat barrier is presented. It is assumed that the shell is thin under the low speed impact. Due to the arbitrary orientation of the shell, while the pre-impact having a linear speed, the postimpact involves rotational and translational speed. Analytical solution for this problem is based on Hertzian theory (Johnson, W., 1972, Impact Strength of Materials, University of Manchester Institute of Science and Technology, Edward Arnold Publication, London) and the Vella’s analysis (Vella et al., 2012, “Indentation of Ellipsoidal and Cylindrical Elastic Shells,” Phys. Rev. Lett., 109, p. 144302) in conjunction with Newtonian method. Due to the nonlinearity and complexity of the impact equation, classical numerical solutions cannot be employed. Therefore, a linearization method is proposed and a closed form solution for this problem is accomplished. The closed form solution facilitates a parametric study of this type of problems. The closed form solution was validated by an explicit finite element method (FEM). Good agreement between the closed form solution and the FE results is observed. Based on the analytical method the maximum total deformation of the shell, the maximum transmitted force, the duration of the contact, and the rotation of the shell after the impact were determined. Finally, it was concluded that the closed form solutions were trustworthy and appropriate to investigate the impact of inclined elastic ellipsoidal shells with an elastic barrier.

References

References
1.
Engin
,
A. E.
,
1969
, “
The Axisymmetric Response of a Fluid Filled Spherical Shell to a Local Radial Impulse—A Model for Head Injury
,”
J. Biomech.
,
2
(3), pp.
325
341
.10.1016/0021-9290(69)90089-X
2.
Heydari
,
M.
, and
Jani
,
S.
,
2010
, “
An Ellipsoidal Model for Studying Response of Head Impacts
,”
Acta. Bioeng. Biomech.
,
12
(
1
), pp.
47
53
.http://www.actabio.pwr.wroc.pl/Vol12No1/8.pdf
3.
Young
,
P. G.
,
2003
, “
An Analytical Model to Predict the Response of Fluid Filled Shells to Impact: A Model for Blunt Head Impacts
,”
J. Sound Vib.
,
267
(5), pp.
1107
1126
.10.1016/S0022-460X(03)00200-1
4.
Vella
,
D.
,
Ajdari
,
A.
, and
Vaziri
,
A.
,
2012
, “
Indentation of Ellipsoidal and Cylindrical Elastic Shells
,”
Phys. Rev. Lett.
,
109
(14), p.
144302
.10.1103/PhysRevLett.109.144302
5.
Reissner
,
E.
,
1947
, “
Stresses and Small Displacements of Shallow Spherical Shells II
,”
J. Math. Phys.
,
25
(
1
), pp.
289
300
.
6.
Ressner
,
E.
,
1959
, “
On the Solution of a Class Problems in Membrane of Thin Shells
,”
J. Mech. Phys. Solids
,
7
(
3
), pp.
242
246
.10.1016/0022-5096(59)90023-7
7.
Kenner
,
V. H.
, and
Goldsmith
,
W.
,
1972
, “
Dynamics Loading of a Fluid Filled Spherical Shell
,”
Int. J. Mech. Sci.
,
14
(9), pp.
557
568
.10.1016/0020-7403(72)90056-2
8.
Kunukkasseril
,
V. X.
, and
Palaninathan
,
R.
,
1975
, “
Impact Experiments on Shallow Spherical Shells
,”
J. Sound Vib.
,
40
(1), pp.
101
117
.10.1016/S0022-460X(75)80233-1
9.
Hammel
,
J.
,
1976
, “
Aircraft Impact on a Spherical Shell
,”
Nucl. Eng. Des.
,
37
(
5
), pp.
205
223
.10.1016/0029-5493(76)90016-9
10.
Senitskii
,
Y. E.
,
1982
, “
Impact of a Viscoelastic Solid Along a Shallow Spherical Shell
,”
Mech. Solids
,
17
(
2
), pp.
120
124
.
11.
Stein
,
E.
, and
Wriggers
,
P.
,
1982
, “
Calculation of Impact-Contact Problems of Thin Elastic Shells Taking Into Account Geometrical Nonlinearities Within The Contact Region
,”
Comput. Methods Appl. Mech. Eng.
,
34
(
9
), pp.
861
880
.10.1016/0045-7825(82)90092-5
12.
Koller
,
M. G.
, and
Busenhart
,
M.
,
1986
, “
Elastic Impact of Spheres on Thin Shallow Spherical Shells
,”
J. Impact Eng.
,
4
(1), pp.
11
21
.10.1016/0734-743X(86)90024-2
13.
Chun
,
M. J.
,
Zhou
,
W. W.
, and
Gui-Tong
,
Y.
,
1992
, “
A Numerical Calculation of Dynamic Buckling of Thin Shallow Spherical Shell Under Impact
,”
J. Appl. Math. Mech.
,
13
(
2
), pp.
125
134
.10.1007/BF02454235
14.
Sabodash
,
P. F.
, and
Zhemkova
,
E. B.
,
1993
, “
Dynamic Reaction of a Spherical Shell Under a Local Normal Load
,”
J. Math. Sci.
,
65
(
6
), pp.
1436
1439
.10.1007/BF01105292
15.
Pauchard
,
L.
, and
Rica
,
S.
,
1998
, “
Contact Compression of Elastic Spherical Shells: The Physics of a Pin Pong Ball
,”
Philos. Mag. B
,
78
(
2
), pp.
225
233
.10.1080/13642819808202945
16.
Gupta
,
N. K.
, and
Venkatesh
,
2004
, “
Experimental and Numerical Studies of Dynamic Axial Compression of Thin Walled Spherical Shells
,”
Int. J. Impact Eng.
,
30
(
8–9
), pp.
1225
1240
.10.1016/j.ijimpeng.2004.03.009
17.
Gupta
,
N. K.
,
Sheriff
,
N. M.
, and
Velmurugan
,
R.
,
2007
, “
Experimental and Numerical Investigations Into Collapse Behavior of Thin Spherical Shells Under Drop Hammer Impact
,”
Int. J. Solids Struct.
,
44
(
10
), pp.
3136
3155
.10.1016/j.ijsolstr.2006.09.014
18.
Her
,
S. C.
, and
Liao
,
C. C.
,
2008
, “
Analysis of Elastic Impact on Thin Shell Structures
,”
Int. J. Mod. Phys. B
,
22
(09n11), pp.
1349
1354
.10.1142/S0217979208046761
19.
Rossikhin
,
Y. A.
,
Shitikova
,
M. V.
, and
Shamarin
,
V.
,
2011
, “
Dynamic Response of Spherical Shells Impacted by Falling Objects
,”
Int. J. Mech.
,
3
(
5
), pp.
166
181
.http://www.naun.org/main/NAUN/mechanics/20-609.pdf
20.
Rossikhin
,
Y. A.
, and
Shitikova
,
M. V.
,
2007
, “
Transient Response of Thin Bodies Subjected to Impact: Wave Approach
,”
Shock Vib. Dig.
,
39
(4), pp.
273
309
.
21.
Mansoor-Baghaei
,
S.
, and
Sadegh
,
A.
,
2011
, “
Elastic Spherical Shell Impacted With an Elastic Barrier: A Closed Form Solution
,”
Int. J. Solids Struct.
,
48
(
22–23
), pp.
3257
3266
.10.1016/j.ijsolstr.2011.07.016
22.
Karagiozova
,
D.
,
Zhang
,
X. W.
, and
Yu
,
T. X.
,
2012
, “
Static and Dynamic Snap-Through Behavior of an Elastic Spherical Shell
,”
Acta Mech. Sin.
,
28
(
3
), pp.
695
710
.10.1007/s10409-012-0065-z
23.
Mansoor-Baghaei
,
S.
, and
Sadegh
,
A.
,
2013
, “
An Analytical Study of Skull Deformation During Head Impacts
,”
ASME
Paper No. IMECE2013-65120.10.1115/IMECE2013-65120
24.
Rossikhin
,
Y. A.
, and
Shitikova
,
M. V.
,
2013
, “
Analysis of Two Colliding Fractionally Damped Spherical Shells in Modeling Blunt Head Impacts
,”
Cent. Eur. J. Phys.
,
11
(
6
), pp.
760
778
.10.2478/s11534-013-0194-4
25.
Bazhenov
,
V. G.
,
Gonik
,
E. G.
,
Kibets
,
A. I.
, and
Shoshin
,
D. V.
,
2014
, “
Stability and Limit States of Elastoplastic Spherical Shells Under Static and Dynamic Loading
,”
ASME J. Appl. Mech. Tech. Phys.
,
55
(
1
), pp.
8
15
.10.1134/S0021894414010027
26.
Mansoor-Baghaei
,
S.
, and
Sadegh
,
A.
,
2015
, “
A Closed Form Solution for the Impact Analysis of Elastic Ellipsoidal Thin Shells
,”
J. Thin Walled Struct.
(in press).
27.
Johnson
,
W.
,
1972
,
Impact Strength of Materials
,
University of Manchester Institute of Science and Technology, Edward Arnold Publication
, London.
28.
Pressley
,
A.
,
2001
,
Elementary Differential Geometry
,
Springer-Verlag
,
London
.
29.
Gerald
,
F.
,
2003
,
Applied Numerical Analysis
,
7th ed.
,
Addison Wesley
, Boston.
You do not currently have access to this content.