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Journal of Pressure Vessel Technology | Volume 132 | Issue 4 | Research Paper
Research Papers: Fluid-Structure Interaction

Linear Stability Analysis and Buckling of Two-Layered Shells Under External Circumferential Loading: A Numerical Investigation

[+] Author and Article Information
George Papadakis

Department of Mechanical Engineering, King’s College London, London WC2R 2LS, U.K.george.papadakis@kcl.ac.uk

J. Pressure Vessel Technol 132(4), 041301 (Jul 21, 2010) (7 pages) doi:10.1115/1.4001638 History: Received October 08, 2009; Revised April 15, 2010; Published July 21, 2010; Online July 21, 2010
Article
References
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The purpose of this paper is to examine computationally the stability of shells consisting of two layers when subjected to external circumferential strain. This loading appears often in biomedicine when the smooth muscle surrounding various organs such as esophagus, lung airways, or gastrointestinal tract contracts. The differential stability equations are discretized using the finite volume method and the resulting generalized eigenvalue problem is solved using the QZ decomposition technique. The predicted number of folds agrees well with available experimental measurements. The present results show that the buckling behavior under circumferential strain loading is entirely different compared with external hydrostatic pressure loading. More specifically, in the latter case, the number of folds with the smallest critical load is always equal to 2. In the former case, however, it depends on the thickness and modulus of elasticity of each layer. The thickness of the inner layer significantly affects the number of folds and the critical buckling load. The influence of the thickness of the outer layer and the ratio of the two moduli of elasticity was also examined, but their effect was not as strong as that of the thickness of the inner layer.
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Copyright © 2010 by American Society of Mechanical Engineers
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References

1
Flügge, W., 1960, "Stresses in Shells", Springer-Verlag, Berlin.
2
Ugural, A. C., 1999, "Stresses in Plates and Shells", 2nd ed., McGraw-Hill, New York.
3
Timoshenko, S. P., 1964, "Theory of Plates and Shells", 2nd ed., McGraw-Hill, New York.
4
Brush, D. O., and Almroth, B. O., 1975, "Buckling of Bars, Plates and Shells", McGraw-Hill, New York.
5
Timoshenko, S. P., and Gere, J. M., 1961, "Theory of Elastic Stability", McGraw-Hill, New York.
6
Kardomateas, G. A., 1993, “Buckling of Thick Orthotropic Cylindrical Shells Under External Pressure,” ASME J. Appl. Mech., 60 , pp. 195–202.
7
Papadakis, G., 2008, “Buckling of Thick Cylindrical Shells Under External Pressure: A New Analytical Expression for the Critical Load and Comparison With Elasticity Solutions,” Int. J. Solids Struct., 45 (20), pp. 5308–5321.  [CrossRef]
8
Bai, A., Eidelman, D. H., Hogg, J. C., James, A. L., Lambert, R. K., Ludwig, M. S., Martin, J., McDonald, D. M., Mitzner, W. A., Okazawa, M., Pack, R. J., Paré, P. D., Schellenberg, R. R., Tiddens, H. A. W. M., Wagner, E. M., and Yager, D., 1994, “Proposed Nomenclature for Quantifying Subdivisions of the Bronchial Wall,” J. Appl. Physiol., 77 (2), pp. 1011–1014.
9
Hrousis, C. A., 1998, “Computational Modelling of Asthmatic Airway Collapse,” Ph.D. thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA.
10
Hrousis, C. A., Wiggs, B. J., Drazen, J. M., Parks, D. M., and Kamm, R. D., 2002, “Mucosal Folding in Biologic Vessels,” J. Biomech. Eng., 124 , pp. 334–341.  [CrossRef]
11
Novozhilov, V. V., 1953, "Foundations of the Nonlinear Elasticity Theory", Dover, New York.
12
Ugural, A. C., and Fenster, S. K., 2003, "Advanced Strength and Applied Elasticity", 4th ed., Prentice-Hall, Englewood Cliffs, NJ.
13
Lai, W. M., Rubin, D., and Krempl, E., 1996, “Introduction to Continuum Mechanics,” 3rd ed., Butterworth, Washington, DC.
14
Versteeg, H. K., and Malalasekera, W., 2007, "An Introduction to Computational Fluid Dynamics: The Finite Volume Method", 2nd ed., Pearson Education Limited, Harlow, England.
15
Pozrikidis, C., 1998, "Numerical Computation in Science and Engineering", Oxford University Press, New York.

Figures

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+Airway wall deformation under imposed circumferential strain eθθ
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Airway wall deformation under imposed circumferential strain eθθ
Figure 1

Airway wall deformation under imposed circumferential strain eθθ

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+Comparison between measurements and predictions for the most unstable mode number: (a) cases 1–12 and (b) cases 13–24
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Comparison between measurements and predictions for the most unstable mode number: (a) cases 1–12 and (b) cases 13–24
Figure 2

Comparison between measurements and predictions for the most unstable mode number: (a) cases 1–12 and (b) cases 13–24

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+Critical load against mode number: (a) circumferential strain loading and (b) hydrostatic pressure loading (ti/R=0.1).
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Critical load against mode number: (a) circumferential strain loading and (b) hydrostatic pressure loading (ti/R=0.1).
Figure 3

Critical load against mode number: (a) circumferential strain loading and (b) hydrostatic pressure loading (ti/R=0.1).

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+Displacement eigen-functions for (a) radial component and (b) circumferential component (pressure loading ti/R=0.1)
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Displacement eigen-functions for (a) radial component and (b) circumferential component (pressure loading ti/R=0.1)
Figure 4

Displacement eigen-functions for (a) radial component and (b) circumferential component (pressure loading ti/R=0.1)

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+Displacement eigen-functions for (a) radial component and (b) circumferential component (circumferential strain loading ti/R=0.1)
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Displacement eigen-functions for (a) radial component and (b) circumferential component (circumferential strain loading ti/R=0.1)
Figure 5

Displacement eigen-functions for (a) radial component and (b) circumferential component (circumferential strain loading ti/R=0.1)

Grahic Jump Location
+Critical load against mode number (a) circumferential strain loading and (b) hydrostatic pressure loading (ti/R=0.05)
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Critical load against mode number (a) circumferential strain loading and (b) hydrostatic pressure loading (ti/R=0.05)
Figure 6

Critical load against mode number (a) circumferential strain loading and (b) hydrostatic pressure loading (ti/R=0.05)

Grahic Jump Location
+Variation in the preferred number of folds against normalized thickness of the internal layer
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Variation in the preferred number of folds against normalized thickness of the internal layer
Figure 7

Variation in the preferred number of folds against normalized thickness of the internal layer

Grahic Jump Location
+Variation in the preferred number of folds against normalized thickness of the outer layer
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Variation in the preferred number of folds against normalized thickness of the outer layer
Figure 8

Variation in the preferred number of folds against normalized thickness of the outer layer

Grahic Jump Location
+Variation in the preferred number of folds against the ratio of moduli of elasticity of the two layers
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Variation in the preferred number of folds against the ratio of moduli of elasticity of the two layers
Figure 9

Variation in the preferred number of folds against the ratio of moduli of elasticity of the two layers

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