Buckling of a Cylinder

Everybody knows how to make a Coke can ready for recycling - step on it, squash it. One also knows that the can resists the squashing up to a certain point, then it suddenly buckles. The can is an example of a thin cylindrical shell. The process of buckling of such shells has not yet been completely understood. One of the main questions is how big is the load that the shell can withstand before it buckles.

Small deformations of thin cylindrical shells are described by the von Karman-Donnell equations. Their normalized form is given by


,

where the nonlinear term is defined as .

Figure 1
Fig. 1 

Function is unknown and describes the radial displacement of a point on the surface of the shell (Figure 1, the quantities displayed in the figure have not been normalized). The distance of the point from the axis of the cylinder is hence , where is the radius of the undeformed cylinder. The auxiliary function can be obtained by solving the second equation for a given . The equations need to be accompanied by suitable boundary conditions (periodic in ). The properties of the cylinder are further described by its thickness and other material constants (eliminated from the equations by the normalization). And finally, represents the load applied on the cylinder.


References

[1]  G. W. Hunt, G. J. Lord, M. A. Peletier, Cylindrical shell buckling: a characterization of localization and periodicity, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 4, 505-518.
[2]  G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. Ahmer Wadee, C. J. Budd, G. J. Lord, Cellular buckling in long structures, Nonlinear Dynam. 21 (2000), no. 1, 3-29.
[3]  J. Horak, G. J. Lord, M. A. Peletier, Cylinder buckling: the mountain pass as an organizing center, SIAM J. Appl. Math. 66 (2006), no. 5, 1793-1824.
[4]  J. Horak, G. J. Lord, M. A. Peletier, Numerical variational methods applied to cylinder buckling, SIAM J. Sci. Comput. 30 (2008), no. 3, 1362-1386.


Updated on March 5, 2009