## Quantified `shock-sensitivity' above the Maxwell load

J.M.T. Thompson & G.H.M. van der Heijden

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_L, 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_L 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 shock 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.
keywords: Maxwell load, shell buckling, shock sensitivity, localization,
imperfection sensitivity, stability, rods, cylindrical constraint

*International Journal of Bifurcation and Chaos* **24**, 1430009
(2014)