Singular Perturbations and Torsional Wrinkling in a Truncated Hemispherical Thin Elastic Shell

Abstract

The work described in this paper is concerned with providing a rational asymptotic analysis for the partial wrinkling bifurcation of a thin elastic hemispherical segment in which the upper rim experiences in-plane circular shearing relative to the other circular edge. The mathematical structure of the associated complex-valued boundary eigenvalue problem is revealed by using the method of matched asymptotic expansions. Our key result is a three-term asymptotic formula for the critical load in terms of a suitable small parameter proportional to the ratio between the thickness and the radius of the shell. Comparisons of this formula with direct numerical simulations provide further insight into the range of validity of the results derived herein.

Self-weight buckling of thin elastic shells: the case of a spherical equatorial segment

“…Standard scaling arguments can help establish the relevant orders of magnitude of these length-scales, but in the interest of brevity such heuristic arguments will not be re-iterated here. We remark in passing that the structure of the eigenmodes is very similar to the situation encountered in our earlier works regarding the wrinkling of flat annular plates [29,38,39] as well as in the recent paper [40] dealing with the azimuthal shearing of a truncated hemispherical elastic shell. With this in mind, we introduce the re-scaled variable Y > 0 defined by…”

“…Taking advantage of the boundary-layer nature of the eigenmodes as α → 0 + , our approach has relied upon the general strategy developed in a number of our previous studies (e.g., [27,29,30,38,40]). However, the case when |k−1| = O(1) turned out to be more subtle than originally anticipated; despite the presence of two distinct branches of eigen-solutions that share the same leading-order term (i.e., an asymptotically double eigenvalue), the original multi-scale ansatz proposed in §5 -see equation (5.2), seems to have "missed" one of the eigen-branches.…”

Lateral buckling of drill strings revisited: localised “snaking”

“…As it happens, this asymptotic structure turns out to be quite different from that already investigated previously in similar contexts (e.g., [18,19,21]) and requires a major change of tack. We note in passing that the behaviour of the critical eigenmodes in the present study shares several salient features in common with our previous work [31,32,33,34,35] dealing with the wrinkling of various thin plate or shell configurations. The paper concludes in §5 with a brief discussion of the main results and their extensions to related situations.…”

“…We shall designate these cases as β << β * and β >> β * , respectively (although, strictly speaking, this is an abuse of notation). We note that the solution samples in Figures 3 and 4 resemble closely some of those encountered in our previous studies [31–33, 35], an observation which suggests that the asymptotic strategy developed therein might be relevant to the boundary-value equations (6)–(8) as well. This is confirmed next.…”

“…The O(μ −1/3 ) contribution of equation ( 13) corresponds to the function V 2 (Z(Y )) cos(μ 1/6 Y ). To satisfy the requisite boundary conditions at Y = 0 (at leading-order, at least) requires that V 2 Z=−ζ 0 = 0; then, a simple use of equation (32) produces:…”