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Rigorous Derivation of the Formula for the Buckling Load in Axially Compressed Circular Cylindrical Shells

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Abstract

The goal of this paper is to apply the recently developed theory of buckling of arbitrary slender bodies to a tractable yet non-trivial example of buckling in axially compressed circular cylindrical shells, regarded as three-dimensional hyperelastic bodies. The theory is based on a mathematically rigorous asymptotic analysis of the second variation of 3D, fully nonlinear elastic energy, as the shell’s thickness goes to zero. Our main results are a rigorous proof of the classical formula for buckling load and the explicit expressions for the relative amplitudes of displacement components in single Fourier harmonics buckling modes, whose wave numbers are described by Koiter’s circle. This work is also a part of an effort to understand the root causes of high sensitivity of the buckling load of axially compressed cylindrical shells to imperfections of load and shape.

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Notes

  1. The assumption that the reference configuration is stress-free is also essential.

  2. While hyperelasticity is hardly the “ultimate” theory of elasticity, it is sufficiently general to permit rigorous study of the asymptotics of the buckling load of a slender structure.

  3. A deformation y is called a weak local minimizer, if it delivers the smallest value of the energy \(\mathcal {E}(\boldsymbol{y})\) among all Lipschitz function satisfying boundary conditions (2.1) that are sufficiently close to y in the W 1,∞ norm.

  4. We restrict our attention to Lipschitz equilibria for technical simplicity of the theory. On the one hand, in most cases of interest, and for a cylindrical shells in particular, the strains and stresses in the trivial branch are uniformly bounded. On the other, the presence or absence of higher spatial derivatives of strains and stresses are immaterial for the theory, and hence no assumptions about them are made.

  5. The set \({\mathcal{A}}_{h}\) could be empty, in which case there are no destabilizing variations, so that the trivial branch remains stable in a neighborhood of (0,0) in the (h,λ)-plane.

  6. Here we consider cylindrical shells of radius 1 and lengths that are uniformly bounded away from 0 and infinity. We therefore do not examine explicit dependence of our constants on L, since it does not affect our results. The cases L→∞ or L→0 are beyond the scope of this paper.

  7. This lemma highlights the fact that Part (b) in Definition 2.10 is designed to capture a buckling mode. We make no attempt to characterize an infinite set of geometrically distinct, yet energetically equivalent buckling modes that exist in our example.

  8. Meaning that each component of e(ϕ) and its trace either changes sign or remains unchanged.

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Acknowledgements

The authors are grateful to Eric Clement and Mark Peletier for their valuable comments and suggestions. This material is based upon work supported by the National Science Foundation under Grant No. 1412058.

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Authors and Affiliations

  1. Temple University, Philadelphia, PA, USA

    Yury Grabovsky

  2. University of Utah, Salt Lake City, UT, USA

    Davit Harutyunyan

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  1. Yury Grabovsky
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  2. Davit Harutyunyan
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Correspondence to Yury Grabovsky.

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Grabovsky, Y., Harutyunyan, D. Rigorous Derivation of the Formula for the Buckling Load in Axially Compressed Circular Cylindrical Shells. J Elast 120, 249–276 (2015). https://doi.org/10.1007/s10659-015-9513-x

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