0
Research Papers

Buckling of a Pressurized Hemispherical Shell Subjected to a Probing Force

[+] Author and Article Information
Joel Marthelot

Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology, Cambridge,
MA 02139

Francisco López Jiménez

Ann and H.J. Smead Department of Aerospace
Engineering Sciences,
University of Colorado,
Boulder, CO 80309

Anna Lee

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

John W. Hutchinson

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

Pedro M. Reis

Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology, Cambridge,
MA 02139;
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 31, 2017; final manuscript received September 26, 2017; published online October 19, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(12), 121005 (Oct 19, 2017) (9 pages) Paper No: JAM-17-1475; doi: 10.1115/1.4038063 History: Received August 31, 2017; Revised September 26, 2017

We study the buckling of hemispherical elastic shells subjected to the combined effect of pressure loading and a probing force. We perform an experimental investigation using thin shells of nearly uniform thickness that are fabricated with a well-controlled geometric imperfection. By systematically varying the indentation displacement and the geometry of the probe, we study the effect that the probe-induced deflections have on the buckling strength of our spherical shells. The experimental results are then compared to finite element simulations, as well as to recent theoretical predictions from the literature. Inspired by a nondestructive technique that was recently proposed to evaluate the stability of elastic shells, we characterize the nonlinear load-deflection mechanical response of the probe for different values of the pressure loading. We demonstrate that this nondestructive method is a successful local way to assess the stability of spherical shells.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Koiter, W. T. , 1966, “ On the Nonlinear Theory of Thin Elastic Shells,” Proc., Ser. B, 69(1 ), pp. 1–54.
Koiter, W. T. , 1967, “ General Equations of Elastic Stability for Thin Shells,” Symposium on the Theory of Shells to Honor Lloyd Hamilton Donnett, pp. 187–230.
Lee, A. , López Jiménez, F. , Marthelot, J. , Hutchinson, J. W. , and Reis, P. M. , 2016, “ The Geometric Role of Precisely Engineered Imperfections on the Critical Buckling Load of Spherical Elastic Shells,” ASME J. Appl. Mech., 83(11), p. 111005. [CrossRef]
Zoelly, R. , 1915, “ Ueber Ein Knickungsproblem an Der Kugelschale,” Ph.D. thesis, ETH Zürich, Zürich, Switzerland.
Samuelson, L. Å. , and Eggwertz, S. F. , 1992, Shell Stability Handbook, Elsevier Applied Science, London.
Thompson, J. M. T. , and Van der Heijden, G. , 2014, “ Quantified ‘Shock-Sensitivity’ Above the Maxwell Load,” Int. J. Bif. Chaos, 24(3), p. 1430009. [CrossRef]
Hutchinson, J. W. , 2016, “ Buckling of Spherical Shells Revisited,” Proc. R. Soc. A, 472(2195), p. 20160577. [CrossRef]
López Jiménez, F. , Marthelot, J. , Lee, A. , Hutchinson, J. W. , and Reis, P. M. , 2017, “ Technical Brief: Knockdown Factor for the Buckling of Spherical Shells Containing Large-Amplitude Geometric Defects,” ASME J. Appl. Mech., 84(3), p. 034501. [CrossRef]
Lee, A. , Brun, P.-T. , Marthelot, J. , Balestra, G. , Gallaire, F. , and Reis, P. M. , 2016, “ Fabrication of Slender Elastic Shells by the Coating of Curved Surfaces,” Nat. Commun., 7, p. 11155. [CrossRef] [PubMed]
Thompson, J. M. T. , 2015, “ Advances in Shell Buckling: Theory and Experiments,” Int. J. Bif. Chaos, 25(1), p. 1530001. [CrossRef]
Thompson, J. M. T. , and Sieber, J. , 2016, “ Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling,” Int. J. Bif. Chaos, 26(2), p. 1630003. [CrossRef]
Virot, E. , Kreilos, T. , Schneider, T. , and Rubinstein, S. , 2017, Stability Landscape of Shell Buckling (submitted).
Hutchinson, J. W. , and Thompson, J. M. T. , 2017, “ Nonlinear Buckling Interaction for Spherical Shells Subject to Pressure and Probing Forces,” ASME J. Appl. Mech., 84(6), p. 061001. [CrossRef]
Carlson, R. L. , Sendelbeck, R. L. , and Hoff, N. J. , 1967, “ Experimental Studies of the Buckling of Complete Spherical Shells,” Exp. Mech., 7(7), pp. 281–288. [CrossRef]
Babcock, C. D. , 1983, “ Shell Stability,” ASME J. Appl. Mech., 50(4b), pp. 935–940. [CrossRef]
Kreilos, T. , and Schneider, T. M. , 2017, “ Fully Localized Post-Buckling States of Cylindrical Shells Under Axial Compression,” Proc. R. Soc. A, 473(2205), p. 20170177.
Evkin, A. Y. , and Lykhachova, O. V. , 2017, “ Energy Barrier as a Criterion for Stability Estimation of Spherical Shell Under Uniform External Pressure,” Int. J. Solids Struct., 118–119, pp. 14–23. [CrossRef]
Lazarus, A. , Florijn, H. , and Reis, P. , 2012, “ Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells,” Phys. Rev. Lett., 109(14), p. 144301. [CrossRef] [PubMed]
Hutchinson, J. W. , and Thompson, J. M. T. , 2017, “ Nonlinear Buckling Behaviour of Spherical Shells: Barriers and Symmetry-Breaking Dimples,” Phil. Trans. R. Soc. A, 375(2093), p. 20160154. [CrossRef]
Nasto, A. , Ajdari, A. , Lazarus, A. , Vaziri, A. , and Reis, P. M. , 2013, “ Localization of Deformation in Thin Shells Under Indentation,” Soft Matter, 9(29), pp. 6796–6803. [CrossRef]
Nasto, A. , and Reis, P. M. , 2014, “ Localized Structures in Indented Shells: A Numerical Investigation,” ASME J. Appl. Mech., 81(12), p. 121008. [CrossRef]
Riks, E. , 1979, “ An Incremental Approach to the Solution of Snapping and Buckling Problems,” Int. J. Solids Struct., 15(7), pp. 529–551. [CrossRef]
Marthelot, J. , Brun, P.-T. , López Jiménez, F. , and Reis, P. M. , 2017, “ Reversible Patterning of Spherical Shells Through Constrained Buckling,” Phys. Rev. Mater., 1(2), p. 025601. [CrossRef]
Koga, T. , and Hoff, N. J. , 1969, “ The Axisymmetric Buckling of Initially Imperfect Complete Spherical Shells,” Int. J. Solids Struct., 5(7), pp. 679–697. [CrossRef]
Reissner, E. , 1946, “ Stresses and Small Displacements of Shallow Spherical Shells. II,” J. Math. Phys., 25(1–4), pp. 279–300. [CrossRef]
Audoly, B. , and Pomeau, Y. , 2010, Elasticity and Geometry: From Hair Curls to the Non-Linear Response of Shells, Oxford University Press, Oxford, UK.
Pogorelov, A. V. , 1988, Bendings of Surfaces and Stability of Shells, Vol. 72, AMS Bookstore, Providence, RI.
Gomez, M. , Moulton, D. E. , and Vella, D. , 2016, “ The Shallow Shell Approach to Pogorelov’s Problem and the Breakdown of Mirror Buckling,” Proc. R. Soc. A, 472, p. 20150732. [CrossRef]
Hutchinson, J. W. , and Thompson, J. M. T. , 2017, “ Imperfections and Energy Barriers in Shell Buckling,” Int. J. Solids Struct. (submitted). https://www.researchgate.net/publication/319471962_Imperfections_and_Energy_Barriers_in_Shell_Buckling
Sanders, J. L., Jr. , 1963, “ Nonlinear Theories for Thin Shells,” Quart. Appl. Math., 21(1), pp. 21–36. [CrossRef]
Evkin, A. , Kolesnikov, M. , and Prikazchikov, D. A. , 2016, “ Buckling of a Spherical Shell Under External Pressure and Inward Concentrated Load: Asymptotic Solution,” Math. Mech. Solids, 22(6), pp. 1425–1437.

Figures

Grahic Jump Location
Fig. 1

(a) Schematic diagram of the fabrication of a shell with a frozen dimple imperfection with a defect amplitude δ. (b) Sketch of the precisely imperfect shell of thickness t, characterized by a defect amplitude δ and a defect angle β. The shell is probed by a indenter of radius a at an angle θ imposing an additional deflection ξ. (c) Photograph of the experimental setup used to measure the pressure variation of a thin hemispherical shells, onto which we impose a set indentation displacement using a universal testing machine (Instron).

Grahic Jump Location
Fig. 2

Knockdown factor, κd, versus indentation depth, ξ¯. In the experiments (closed symbols), the shells (R = 24.9 mm, t = 220–250 μm) were fabricated with initial normalized defect amplitude δ¯=0.26,0.40,0.60,0.85. The solid lines represent simulations by FEM for the corresponding defect profile. The dashed lines are the numerical prediction for a frozen defect with the same geometry.

Grahic Jump Location
Fig. 3

(a) Knockdown factor, κd, versus the indentation depth, ξ¯, calculated from FEM, for a generic Gaussian dimple with a profile defined by Eq. (3). The radius-to-thickness ratio of the shells is R/t = 100, containing Gaussian dimples with λ = 3 and δ¯=[0.05, 0.1, 0.2, …, 0.9, 1], distributed between 0.1 and 1 by step of 0.1. (b) Critical displacement, ξ¯c, corresponding to a knockdown factor of 0.95κd0 versus the amplitude of the initial imperfection, δ¯, for shells with λ = [2.615, 3, 4, 5, 10] and an indenter radius a = 200 μm.

Grahic Jump Location
Fig. 4

Experimental knockdown factor, κd, versus indentation depth, ξ¯, for four geometries of the indenter characterized by their curvature radius (a = [0.2, 4, 22, ] mm) for the same shell (R = 24.9 mm, t = 210 μm) fabricated with an initial normalized defect amplitude δ¯=0.62. (b) Knockdown factor, κd(a = 22), computed by FEM with an indenter of radius a = 22 mm normalized by the knockdown factor, κd(a = 0.22) computed considering an indenter of radius a = 0.22 mm for a generic Gaussian dimple with a profile defined by Eq. (3). The radius-to-thickness ratio is R/t = 100 and the shells contain Gaussian dimples with λ = 2.615.

Grahic Jump Location
Fig. 5

Normalized probe force, FR/2πD, versus normalized indentation depth, ξ¯, for a shell (R = 24.9 mm, t = 210 μm, κd0 = 0.74) subjected to 11 prescribed internal pressures. Each curve is the mean of six independent, but otherwise identical, force–displacement curves. Solid lines are predictions from Ref. [13].

Grahic Jump Location
Fig. 6

Normalized maximum probe force, FmaxR/2πD, versus prescribed internal pressure normalized by the critical pressure, po/pc, for a shell with R = 24.9 mm, t = 210 μm, and κd0 = 0.74. The solid line is a prediction from Ref. [13] for a perfect shell. (Inset) Raw and filtered force signals versus displacement for a shell subjected to an internal pressure po/pc = 0.21.

Grahic Jump Location
Fig. 7

Normalized energy barrier, W/Wc, versus the normalized internal pressure po/pc for shells with no initial engineered defect (R = 24.9 mm, t = 210 μm, κd0 = 0.74, solid circles) and decreasing value of κd0 (R = 24.9 mm, t = 210–230 μm, κd0 = [0.53, 0.43, 0.22]). Solid lines are predictions from Ref. [13], for a perfect shell and dashed lines specialized for an initial Gaussian dimple with β0 = 10.8 deg and δ¯=0.62. (Inset) Log–log plot of the same data.

Grahic Jump Location
Fig. 8

(a) Variation of the knockdown factor, κd, versus probe displacement, ξ¯, for a shell (R = 24.9 mm, t = 210 μm) containing an initial an imperfection with δ¯=0.62. The angle θ is defined between the direction perpendicular to the defect at the shell pole and the direction of indentation. (Inset) Maximum knockdown factor normalized by κd0 versus angle of indentation normalized by the defect angle β0. (b) Variation of the energy barrier with the normalized internal pressure po/pc for two indentation angle θ for the same shell (R = 24.9 mm, t = 210 μm, κd0 = 0.43).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In