Research Papers

Nonlinear Buckling Interaction for Spherical Shells Subject to Pressure and Probing Forces

[+] Author and Article Information
John W. Hutchinson

School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138

J. Michael T. Thompson

Department of Applied Maths and
Theoretical Physics,
University of Cambridge,
Cambridge CB3 0WA, UK

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 9, 2017; final manuscript received March 22, 2017; published online April 12, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(6), 061001 (Apr 12, 2017) (11 pages) Paper No: JAM-17-1135; doi: 10.1115/1.4036355 History: Received March 09, 2017; Revised March 22, 2017

Elastic spherical shells loaded under uniform pressure are subject to equal and opposite compressive probing forces at their poles to trigger and explore buckling. When the shells support external pressure, buckling is usually axisymmetric; the maximum probing force and the energy barrier the probe must overcome are determined. Applications of the probing forces under two different loading conditions, constant pressure or constant volume, are qualitatively different from one another and fully characterized. The effects of probe forces on both perfect shells and shells with axisymmetric dimple imperfections are studied. When the shells are subject to internal pressure, buckling occurs as a nonaxisymmetric bifurcation from the axisymmetric state in the shape of a mode with multiple circumferential waves concentrated in the vicinity of the probe. Exciting new experiments by others are briefly described.

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Fig. 1

(a) Geometry and loads. The influence with p=0 of the width of the rigid disk insert at each pole to which P is applied. The half-pole angle of the disk is β0. (b) Normalized pole force versus normalized pole displacement. Volume change contribution in (c) versus normalized pole displacement. The curves have been computed assuming axisymmetric deformations with R/t=200 and ν=0.3.

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Fig. 2

Variation of the inward radial deflection of the shell, w/R, in (a) and the rotation of the shell middle surface, φ, in (b) for the shell in Fig. 1 with no pressure loading (p=0) and with R/t=200 and ν=0.3 (β0=0.2deg)

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Fig. 3

(a) Illustration of the difference between the responses of the complete spherical shell subject to concentrated forces P at the poles for one case in which the net external pressure p is held constant and the other case where the volume in the shell is constrained to be constant. Both cases have p/pC=ΔV/ΔVC=0.3 at the onset of the application of P. (b) applies to constant p with H specifying the volume change as defined in the text. (c) applies to constant ΔV showing the variation of the net external pressure acting on the shell. These results have been computed with R/t=200 and ν=0.3 but they are essentially independent of R/t and ν as discussed in the text.

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Fig. 4

Dimensionless plots for F in (a) and H in (b) for spherical shells subject to prescribed external pressure. These results have been computed with R/t=200 and ν=0.3 but are essentially independent of R/t and ν.

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Fig. 5

For prescribed external pressure p : (a) maximum probe force. The solid dots are from Table 1, formula (31), in Ref. [10]; (b) energy barrier per pole to buckling by the probe force. These results have been computed with R/t=200 and ν=0.3 but are essentially independent of R/t and ν.

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Fig. 6

Imperfection-sensitivity of spherical shells with dimple imperfections subject to external pressure alone. (a) Pressure versus pole deflection for shells with R/t=200, ν=0.3 and B=1.5. (b) Maximum pressure versus imperfection amplitude for three values of R/t.

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Fig. 7

Application of probe force P at fixed external pressure for a shell with imperfection amplitude δ/t=0.25 and B=1.5. (a) Four pressures identified by dots at which probe force is applied. (b) Relation of P to additional pole deflection for each of the four pressures.

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Fig. 8

Maximum probe force in (a) and energy barrier to buckling in (b) for imperfect spherical shells at prescribed external pressures below pmax. Here, pmax is the buckling pressure for agiven imperfection amplitude δ (with B=1.5) plotted in Fig.6(b).

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Fig. 9

Response of the spherical shell under prescribed internal pressure (p<0) subject to pole force P. These results have been computed with R/t=200 and ν=0.3 but are essentially independent of R/t and ν.

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Fig. 10

The effect of reducing the bending stiffness of the shell to DM on the relation of P to pole deflection while keeping the stretching stiffness unchanged. The three levels of internal pressure are: (a) p/pC=−0.5, (b) p/pC=−1, and (c) p/pC=−10. These results have been computed with R/t=200, ν=0.3 and β0=1deg. The bending stiffness in the ordinate is the full bending stiffness, D=Et3/[12(1−ν2)].

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Fig. 11

Probe force versus pole displacement in (a) and associated net external pressure in (b) for spherical shells subject to fixed change in internal volume, ΔV/ΔVC. The net external pressure prior to application of P is (p/pC)ξ=0=ΔV/ΔVC. These results have been computed with R/t=200 and ν=0.3.

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Fig. 12

For prescribed change in volume with ΔV/ΔVC>0 : (a) maximum probe force; (b) energy barrier per pole to buckling. The lower limit ΔVL/ΔVC for which a buckled state exists with P=0 depends on R/t and ν. The normalized maximum probe force Pmax is nearly independent of R/t and ν except for the lower limit ΔVL/ΔVC. There is a slight dependence on R/t and ν for the normalized energy barrier in (b) where the lower limit is evident.

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Fig. 13

Force versus pole displacement in (a) and associated net external pressure in (b) for spherical shells subject to fixed internal volume, ΔV/ΔVC, during application of P generating a net internal pressure. These results have been computed with R/t=200 and ν=0.3.

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Fig. 14

Nonaxisymmetric buckling from the axisymmetric state due to application of pole forces for the case of prescribed pressure. (a) Curve of pole force versus pole deflection for axisymmetric deformation with solid dots indicating the first nonaxisymmetric bifurcation. (b) The value of normalized pole deflection ξ associated with the first bifurcation is plotted as a function of p/pC along with the number of circumferential waves m associated with the critical mode. Note that the scale of the horizontal axis changes by a factor of 100 at p=0. These results have been computed with R/t=500, v=0.3 and c0=0.0482, but to a good approximation they are independent of R/t and ν.

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Fig. 15

An illustration of the nonaxisymmetric bifurcation mode and the associated axisymmetric state for a spherical shell with internal pressure p/pC=−3 and subject to pole forces. Distributions associated with the axisymmetric state in (a) and (b) at bifurcation, and the meridional variation of the critical bifurcation mode in (c) having m=5. The meridional distance from the is pole measured by s¯=s(1−ν2/Rt)1/2. These distributions have been computed with R/t=500, ν=0.3 and c0=0.0482, but they are essentially independent of R/t.

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Fig. 16

Buckling of an inflated rubber ball subject to a cylindrical indenter (Reproduced with permission from Vella et al. [18]. Copyright 2015 by IOPscience.)

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Fig. 17

The results of Figs. 4(a) and 11(a) reproduced in simplified form to aid the present discussion

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Fig. 18

Experiments by Virot, Rubinstein, Kreilos and Schneider on an axially compressed cylindrical shell (the ubiquitous coke can) where both the end load and probe are under rigid displacement-control. The probe’s load–deflection characteristics are shown on the graph for different values of the axial load, and succeed in locating the “free” equilibrium states of the shell, A, where the probing force has dropped to zero.



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