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From:
J. Michael T. Thompson (Department of Applied Maths & Theoretical Physics, University of Cambridge, CB3 0WA, UK), “Advances in Shell Buckling: Theory and Experiments”, http://arxiv.org/abs/1409.3156 , Cornell University Library, Nonlinear Sciences, Pattern Formation and Solitons, (Submitted on 10 Sep 2014), published in the International Journal of Bifurcation and Chaos, Vol. 25, No. 1 (2015) 153001 (25 pages), DOI: 10.1142/S0218127415300013 .
ABSTRACT: In a recent feature article in this journal, co-authored by Gert van der Heijden, I described the static-dynamic analogy and its role in understanding the localized post-buckling of shell-like structures, looking exclusively at integrable systems. We showed the true significance of the Maxwell energy criterion load in predicting the sudden onset of 'shock sensitivity' to lateral disturbances. The present paper extends the survey to cover non-integrable systems, such as thin compressed shells. These exhibit spatial chaos, generating a multiplicity of localized paths (and escape routes) with complex snaking and laddering phenomena. The final theoretical contribution shows how these concepts relate to the response and energy barriers of an axially compressed cylindrical shell. After surveying NASA's current shell-testing programme, a new non-destructive technique is proposed to estimate the 'shock sensitivity' of a laboratory specimen that is in a compressed meta-stable state before buckling. A probe is used to measure the nonlinear load-deflection characteristic under a rigidly applied lateral displacement. Sensing the passive resisting force, it can be plotted in real time against the displacement, displaying an equilibrium path along which the force rises to a maximum and then decreases to zero: having reached the free state of the shell that forms a mountain-pass in the potential energy. The area under this graph gives the energy barrier against lateral shocks. The test is repeated at different levels of the overall compression. If a symmetry-breaking bifurcation is encountered on the path, computer simulations show how this can be supressed by a controlled secondary probe tuned to deliver zero force on the shell.
This is Fig. 21 of the paper. Figure caption is:
Fig. 21 An impression of the proposed experimental procedure in which a rigid probe is used to construct a lateral-load versus lateral-displacement graph, Q(q). This graph ends with Q = 0 at a free equilibrium state of the shell, and the area under the curve gives the corresponding energy barrier. Note that at Q = 0 the rigid probe is stabilizing a state that would otherwise be unstable for the free shell.
Thompson writes:
"Here we have the controlled displacement, q, producing a passive reactive force from the shell, Q, which is sensed by the device, giving finally the plot of Q(q). This is all to be done at a prescribed value of the axial compressive load, P, which might itself be applied in either a dead or rigid manner. In the simplest scenario, the Q(q) graphs might be expected to look like those sketched on the right-hand side, the top for P greater than the Maxwell load [PM] and the lower one for P < PM."
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