The Maxwell tipping point as an organizing centre for unstable collapse

When a long structure buckles unstably under conditions of controlled end-displacement it tends to localize, with deflection limited to only a portion of the length. If the structure then locks-up and restabilizes, localized buckling shifts to a second region along the length, often adjacent to the first. As this process develops into further regions, the load typically tends to fluctuate. This process has been termed cellular buckling and the associated set of equilibrium states, fluctuating in load as the corresponding shortening progresses, are known as a snaking equilibrium path [1].

Such behaviour occurs, for example, in a long thin compressed cylindrical shell, with a localized row of dimple buckles forming around the circumference and then extending into a second and then further rows [2]. The well-known Maxwell load, where energy levels in the unbuckled and final periodic states have the same potential energy, acts as an organizing centre for this process. As end-shortening increases, the load fluctuates about the Maxwell load.

One major characteristic of shells, seen for example in the cylinder under end compression, is that they exhibit “snapback” behaviour; even under conditions of controlled end-shortening rather than load, as is usual in experiments, dynamic buckling occurs accompanied by a drop in load-carrying capacity. Under such circumstances, the dual Maxwell displacement rather than load can act as the organizing centre. The equilibrium path now fluctuates in end-shortening rather than load, which continues to drop as buckling progresses. For the cylinder problem this process describes a buckle pattern that starts with a single dimple, and then grows by introducing further localized dimples in turn around the circumference, as the end-displacement fluctuates back and forth.

Much of the resulting equilibrium path is unstable even under rigid loading, so the pattern may only be observable numerically [3], but it can nevertheless be of significant practical importance. To help describe such complex behaviour, a simple finite degree-of-freedom cellular model is introduced, based on a simple unit cell with a snapback characteristic. This indicates that there are two typical but fundamentally different variants to the behaviour. Each cell buckles with its own Maxwell displacement, and as the buckling develops these can either be separated in terms of evolution under controlled displacement, or overlap. The difference in the presence of background disturbance can be crucial; if buckling is assumed to always to occur at the next Maxwell displacement, the former would allow the system to settle into the different localized states, whereas the latter means that the entire pattern triggers immediately as a domino effect. This change-over point is what we term the Maxwell tipping point.

[1] G.W.Hunt, M.A.Peletier, A.R.Champneys, P.D.Woods, M.A.Wadee, C.J.Budd and G.J.Lord. Cellular buckling in long structures. Nonlinear Dynamics, 21(1):3-29, 2000.
[2] G.J.Lord, A.R.Champneys and G.W.Hunt. Computation of homoclinic orbits in partial differential equations: an application to cylindrical shell buckling. SIAM J.Sci. Comp., 21(2):591-619, 1999.
[3] A.R.Champneys, T.J.Dodwell, R.M.J.Groh, G.W.Hunt, R.M.Neville, A.Pirrera, A.H.Sakhaei, M.Schenk and M.A.Wadee. Happy catastrophe: recent progress in analysis and exploitation of elastic instability. Front. Appl. Math. Stat., 2019. https://doi.org/10.3389/fams.2019.00034