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Research Papers

The Geometric Role of Precisely Engineered Imperfections on the Critical Buckling Load of Spherical Elastic Shells

[+] Author and Article Information
Anna Lee

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

Francisco López Jiménez, Joel Marthelot

Department of Civil and
Environmental 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 Mechanical Engineering,
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: preis@mit.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 4, 2016; final manuscript received August 7, 2016; published online September 1, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(11), 111005 (Sep 01, 2016) (11 pages) Paper No: JAM-16-1386; doi: 10.1115/1.4034431 History: Received August 04, 2016; Revised August 07, 2016

We study the effect of a dimplelike geometric imperfection on the critical buckling load of spherical elastic shells under pressure loading. This investigation combines precision experiments, finite element modeling, and numerical solutions of a reduced shell theory, all of which are found to be in excellent quantitative agreement. In the experiments, the geometry and magnitude of the defect can be designed and precisely fabricated through a customizable rapid prototyping technique. Our primary focus is on predictively describing the imperfection sensitivity of the shell to provide a quantitative relation between its knockdown factor and the amplitude of the defect. In addition, we find that the buckling pressure becomes independent of the amplitude of the defect beyond a critical value. The level and onset of this plateau are quantified systematically and found to be affected by a single geometric parameter that depends on both the radius-to-thickness ratio of the shell and the angular width of the defect. To the best of our knowledge, this is the first time that experimental results on the knockdown factors of imperfect spherical shells have been accurately predicted, through both finite element modeling and shell theory solutions.

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References

Figures

Grahic Jump Location
Fig. 4

Knockdown factor, κd=pmax/pc, versus the normalized defect amplitude, δ¯=δ/t. In experiments (closed symbols), the shell specimens were fabricated in the ranges of parameters, tmold = {585, 975, 1170} μm and 0≤δ¯≤2.36. The lines represent FEM data in which the defect profiles obtained by simulations with tmold = {585, 975, 1170} μm were introduced to vary the angular width of the defect.

Grahic Jump Location
Fig. 3

(a) Profiles of the indented mold calculated by FEM with tmold = 585 μm and 30≤δ (μm)≤300 (in steps of 30 μm) are plotted in (x, y)-coordinates. Inset: Magnified profiles at the vicinity of the pole. (b) Angular profile of the defect versus zenith angle for shells with δ = 207 μm: experiments with tmold={585, 1170} μm (solid lines) and FEM with tmold = {585, 975, 1170} μm (dashed, dashed-dotted-dotted, and dashed-dotted lines, respectively).

Grahic Jump Location
Fig. 2

Fabrication of the thin shell specimens. (a) Photographs and (b) schematic diagrams of the fabrication protocol used to produce thin spherical shells with a dimplelike defect. (1) A thick VPS mold shell is filled with liquid VPS and (2) turned upside down. (3) A dimplelike defect is introduced by indenting the pole of the mold shell with an Instron machine, immediately after pouring of VPS. ((4) and (5)) Upon curing, a thin elastic shell containing a geometric defect is peeled off from the mold.

Grahic Jump Location
Fig. 1

Experimental results of the knockdown factor, κd, versus the radius-to-thickness ratio, η=R/t, of spherical shells. Most of the previous experiments [59] (open symbols) were conducted with shallow spherical segments and resulted in a large variation in κd = 0.17–0.9. Carlson et al. [10] used complete spherical shells and increased the knockdown factor from 0.05 to 0.86 by improving the shell surface and loading conditions. Our near perfect shells (closed circle) have a small variation in κd = 0.61–0.92, which can be lowered significantly by engineering a dimplelike defect (closed square).

Grahic Jump Location
Fig. 6

Knockdown factor, κd, versus the normalized defect amplitude, δ¯, for a variety of λ. (a) Solid lines represent the results of the ODE solutions, and dashed and dotted lines correspond to FEM simulations for different η = {100, 200}, respectively, with 1.5 ≤ λ ≤ 5, in steps of 0.5. (b) FEM results for 1 ≤ λ ≤ 5, in steps of 0.25. The lower bounding envelope (thick solid line) is determined by fitting (Eq. (23)). (c) Critical geometric parameter of the defect, λc, at which κd exhibits its minimum possible value for a given value of δ¯.

Grahic Jump Location
Fig. 5

Comparison between ODE (solid lines) and FEM (dashed lines) of the load-carrying behavior of imperfect shells. (a) Normalized pressure, p¯, as a function of the normalized volume change, V¯. (b) Normalized pressure, p¯, versus the normalized displacement at the pole, w¯. Shells with radius-to-thickness ratio η = 100 containing a Gaussian dimple (Eq. (2)) with β0 = 8.83 deg and δ¯={0.03,0.1,0.3,1} were used.

Grahic Jump Location
Fig. 7

(a) Pressure level of the plateau versus the geometric parameter, λ, of the defect. (b) Normalized defect amplitude at onset of the plateau versus λ. The various values of the threshold, ξ, used to define the plateau are provided in the legend.

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