Localization and solitary waves in solid mechanics
Professor G.W. Hunt
University of Bath
The response of many thin plate and shell structures under compressive loads can be taken as analogous to
that of a long thin strut on a nonlinear Winkler foundation, represented by the ODE
w'''' + p w'' + w + a2w2 +
a3w3 + ... = 0,
where w is lateral displacement, p is applied compression in the direction of the unbuckled
strut, and the ai are constants. Depending on the values of these constants, the strut can
show fundamentally different responses. For example, if a3 >
38a22/27, the post-buckling response mirrors that of a flat plate -- stable,
able to take increasing load, and periodic or quasi-periodic in form. On the other hand, if
a3 < 38a22/27, the initial post-buckling response is
shell-like - unstable, unable to take load beyond the critical bifurcation point, and homoclinic,
connecting the flat (unbuckled) state to itself: then, localized solitary waves would be found in practical
circumstances, although periodic solutions also co-exist. Some combinations of the ai can
induce responses that initially destabilize but subsequently restabilize, giving rise to the possibility of
heteroclinic connections between unbuckled and periodic states.
The talk will outline the fundamental differences between these three types of response. While mathematical
and numerical results from a number of different sources and co-workers are relied upon, it is the physical
interpretation that will be emphasized. Thus the heteroclinic form of behaviour will be explained in terms of
a cellular form of buckling, where local buckles occur in sequential fashion, accompanied by fluctuating
load. It will be demonstrated that the classical Maxwell load, at which the global energy minimum swaps from
unbuckled to periodically-buckled state, appears as the organizing centre for this sequence. If time permits,
examples of such instability phenomena from the field of structural geology will be outlined.
