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Fig. 1.3. Part (a) shows a numerical computation of a solution wMP that is a mountain-pass point of the energy Fλ for a load λ = 1.5. We show both the graph of wMP(x,y) as well as its rendering on a cylinder. At wMP there exist two directions in the state space X in which the energy Fλ decreases. By perturbing wMP in these directions and following a gradient flow of Fλ we move away from wMP. In one direction the dimple shrinks to
zero (not shown) and in the opposite it grows in amplitude and extent (figures (b) and (c)).
From:
J. Horák (1), G. J. Lord (2), M. A. Peletier (3)
(1) Universität Köln, Germany
(2) Heriot-Watt University, Edinburgh, United Kingdom
(3) Technische Universiteit Eindhoven, The Netherlands
“Cylinder buckling: the mountain pass as an organizing center”, SIAM J. Appl. Math. Vol. 66 No. 5, 2006, pp. 1793-1824, arXiv:math/0507263 (Cornell University Library), DOI. 10.1137/050635778
ABSTRACT: We revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this is the existence of a mountain pass point for the system. We prove the existence on bounded domains of such solutions for all most all loads and then numerically compute example mountain pass solutions. Numerically the mountain pass solution with lowest energy has the form of a single dimple. We interpret these results and validate the lower bound against some experimental results available in the literature.
This is Fig. 1.3 from the paper.
Caption: Part (a) shows a numerical computation of a solution wMP that is a mountain pass point of the energy Fλ for a load λ = 1.5. We show the graph of the displacement wMP(x,y) as a function of (x,y) as well as its rendering on a cylinder. At wMP there exist two directions in the state space X in which the energy Fλ decreases. By perturbing wMP in these directions and following a gradient flow of Fλ, we move away from wMP. In one direction the dimple shrinks and disappears (not shown) and in the opposite its direction grows in amplitude and extent ((b) and (c)).
The key result is the existence of a mountain pass point, an equilibrium state that is straddled between two valleys in the energy landscape; one valley surrounds the unbuckled state, and the other contains many buckled, large-deformation states.
This mountain pass point has a number of interesting properties, as follows:
1. It has the appearance of a single-dimple solution, a small buckle in the form of a single dent (see Figure 1.3(a)). Single-dimple deformations have appeared in engineering literature in a number of different ways (see section 6), but a theoretical understanding of this phenomenon is still lacking. Localization (concentration) of deformation is commonly known to appear in extended structures [21], and in the cylinder localization in the axial direction has been studied theoretically and numerically [20, 24, 25]. Whether localization is possible in the tangential direction has been an open problem for some time; it is interesting that our simulations for the perfect structure show solutions that are localized in both axial and tangential directions.
2. Like all mountain pass points, this single-dimple solution is unstable, in the sense that there are directions in state space in which the energy decreases. In one direction the dimple roughly shrinks and disappears, and in the oppo- site direction it grows and multiplies (Figures 1.3(b)–(c)). It is remarkable, however, that our numerical results indicate that the single-dimple solution has an alternative characterization as a constrained global minimizer (a global minimizer of the strain energy under prescribed end shortening).
3. The equations can be rescaled so that the only remaining parameters are the load level and the domain. The geometry of the mountain pass solution we calculate even appears to be independent of the domain size.
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