Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling

Abstract

Under increasing compression, an unbuckled shell is in a metastable state which becomes increasingly precarious as the buckling load is approached. So to induce premature buckling, a lateral disturbance will have to overcome a decreasing energy barrier which reaches zero at buckling. Two archetypal problems that exhibit a severe form of this behavior are the axially-compressed cylindrical shell and the externally pressurized spherical shell. Focusing on the cylinder, a nondestructive technique was recently proposed to estimate the “shock-sensitivity” of a laboratory specimen using a lateral probe to measure the nonlinear load-deflection characteristic. If a symmetry-breaking bifurcation is encountered on the path, computer simulations showed how this can be suppressed by a controlled secondary probe. Here, we extend our understanding by assessing in general terms how a single control can capture remote saddle solutions: in particular, how a symmetric probe could locate an asymmetric solution. Then, more specifically, we analyze the spherical shell with point and ring probes, to test the procedure under challenging conditions to assess its range of applicability. Rather than a bifurcation, the spherical shell offers the challenge of a destabilizing fold (limit point) under the rigid control of the probe.

Buckling of a Pressurized Hemispherical Shell Subjected to a Probing Force

“…For these numerical results [13], the shape of the imperfection was approximated and modeled as a Gaussian dimple (see Eq. (3) As proposed recently [10][11][12], our results demonstrate that measuring the variation of the energy barrier with the normalized internal pressure can be interpreted as a nondestructive technique to probe the stability of a shell that is uniformly compressed close, but just prior, to its working load. The initial amplitude of the imperfection can therefore be extracted by sequentially increasing the depressurization on the shell and: (i) probing the loaddisplacement curve (with a minimum threshold slightly larger than zero for the force signal to avoid the catastrophic collapse of the structure); (ii) computing the corresponding energy barrier; and (iii) increasing the depressurization on the shell.…”

“…Nonetheless, a detailed a priori knowledge of the geometrical imperfections of the shell is still needed in order to predict the stability of such a structure. In an attempt toward circumnavigating this requirement, a novel nondestructive framework has recently been proposed to probe the stability of uniformly compressed cylindrical shells [10][11][12] and spherical shells [13]. The basis of this approach is to measure the relationship between the nonlinear deflections of an elastic shell that was initially compressed close to its working load, by subjecting it to a point indentation force, which hereon we shall refer to as probing force.…”

Probing Shells Against Buckling: A Nondestructive Technique for Laboratory Testing

“…The basic method as proposed by Thompson [2015] and Thompson & Sieber [2016] can be summarized as follows. We first note that the buckle shape at the lowest energy barrier determined by Horak et al [2006] for an axially loaded cylindrical shell looks remarkably like the small dimple that might be pressed into the cylinder by a researcher's finger.…”

“…The various ways that this simple procedure can fail, by for example reaching a vertical tangency or a bifurcation on the F(D) curve before reaching A, are explored in detail by Thompson & Sieber [2016] focusing on simple models of both cylindrical and spherical shells. In particular, they explore the stabilization of the process by means of a secondary probe that is tuned to provide zero force as illustrated in figure 3.…”

“…We shall be looking more closely at these matters in a later section. (Thompson & Sieber [2016]). The illustration of the eigenvector demonstrates how the lateral F(D) curve can be located beyond the bifurcation point, S, by the introduction of a second rigidly controlled point probe, tuned to provide zero force.…”