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Static and dynamic snap-through behaviour of an elastic spherical shell

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Abstract

The deformation and snap-through behaviour of a thin-walled elastic spherical shell statically compressed on a flat surface or impacted against a flat surface are studied theoretically and numerically in order to estimate the influence of the dynamic effects on the response. A table tennis ball is considered as an example of a thin-walled elastic shell. It is shown that the increase of the impact velocity leads to a variation of the deformed shape thus resulting in larger deformation energy. The increase of the contact force is caused by both the increased contribution of the inertia forces and contribution of the increased deformation energy.

The contact force resulted from deformation/inertia of the ball and the shape of the deformed region are calculated by the proposed theoretical models and compared with the results from both the finite element analysis and some previously obtained experimental data. Good agreement is demonstrated.

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Abbreviations

A j :

Parts of the surface area occupied by the regions with a constant meridian curvature, j = 1, 2, ⋯

d, d in :

Outer and inner diameters of the contact area, shown in Fig. 3b

E, ν:

Young’s modulus and Poisson’s ratio

e b, e m :

Bending and membrane strain energy density, respectively

F :

Contact force

h :

Thickness of the shell wall

m(u, r):

Mass of the shell within the deformed part of the shell

p, q :

Proportionality coefficients for the shape parameters

R :

Radius of the spherical shell

R 1, R 2 :

Geometric characteristics of the deformed shell, shown in Fig. 12

r 1, r 2, r 3 :

Characteristic radii of the deformed shell along the meridian

r c, r in :

Current and initial circumferential radii in the deformed regions, respectively

U D :

Deformation energy

u :

Displacement of the shell with respect to the rigid surface

V 0 :

Impact velocity

W E, W IN :

Work done by the external (contact) force and inertia force, respectively

ɛ 1, ɛ 2 :

In-plane strains

K j :

Constant meridian curvature of region j, j = 1, 2, ⋯

κ i , Δκ i :

Curvatures and change of curvatures, respectively, i = 1, 2

ρ :

Material density

ξ(u, r):

Distributed displacements of the deformed region along the direction of loading

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Authors and Affiliations

  1. Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    D. Karagiozova, X. -W. Zhang & T. -X. Yu

  2. Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 4, Sofia, 1113, Bulgaria

    D. Karagiozova

  3. School of Mechantronical Engineering, Beijing Institute of Technology, 100081, Beijing, China

    X. -W. Zhang

Authors
  1. D. Karagiozova
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  2. X. -W. Zhang
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  3. T. -X. Yu
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Correspondence to D. Karagiozova.

Additional information

The project was supported by the National Natural Science Foundation of China (11032001).

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Karagiozova, D., Zhang, X.W. & Yu, T.X. Static and dynamic snap-through behaviour of an elastic spherical shell. Acta Mech Sin 28, 695–710 (2012). https://doi.org/10.1007/s10409-012-0065-z

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