Quantified 'shock sensitivity' above the Maxwell load

Using the static-dynamic analogy, work at Bath and Bristol has uncovered the vital organizing role of the Maxwell 'energy criterion' load in the advanced post-buckling of long-thin structures which exhibit severe shell-like imperfection sensitivity. It has become clear that above the Maxwell load, $P_M$, there are localized solutions offering an order-of-magnitude increase in sensitivity to lateral side-loads, whether static or dynamic. We propose to call this 'shock-sensitivity', and notice that so far only the seminal paper by Horak, Lord and Peletier in 2006 has quantified this in terms of an $E(P)$ energy-barrier versus load graph. In this paper we present three graphs of this nature for archetypal problems: the free twisted rod, the cylindrically constrained rod, and the strut on a softening elastic foundation. We find in all cases that the energy barrier of the localizing solution above $P_M$ is quite close to the energy of a single periodic wave. Now a single such wave is not kinematically admissible, and the corresponding periodic barrier must be for all the waves in the long structure, $N$, say. So in practice $N$ will be large, and does indeed tend to infinity with the length of the structure. Thus the sensitivity increases by a factor of a large $N$ as the Maxwell load is exceeded. This is important in its own right, and we do not seek to explain or fit curves to the scattered experimental buckling loads of shell structures.