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

Abstract

Buckling of an equatorial shallow shell segment under its own weight is re-considered here from a novel point of view involving singular-perturbation techniques. A number of different asymptotic approximations for the critical buckling load are proposed and discussed in detail, while comparisons with direct numerical simulations of the associated boundary-value problem confirm the excellent accuracy of the results obtained.

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 presence of the variable coefficients makes it impossible to solve this analytically (except in special cases), but we can take advantage of the presence of the asymptotic parameter μ 1 to find suitable approximations for its solutions. We remark in passing that the aforementioned boundary eigenvalue problem resembles closely that studied recently in connection with the self-buckling of an equatorial spherical shell segment [31].…”

“…Our interest in the model explored in this work was partially stimulated by the recent publication [31] in which the self-buckling of a shallow spherical equatorial segment was reduced to an equation akin to (6). In that work, we relied on the use of a simplified WKB analysis to deliver highly accurate approximations for the first two lowest eigenvalues of the corresponding bifurcation equation.Although the boundary-layer approach was also tentatively pursued, the full details of the analysis were somewhat glossed over and the application of the boundary conditions for the asymptotic expansion was rather sketchy.…”

“…Our interest in the model explored in this work was partially stimulated by the recent publication [31] in which the self-buckling of a shallow spherical equatorial segment was reduced to an equation akin to (2.6). In that work we relied on the use of a simplified WKB analysis to deliver highly accurate approximations for the first two lowest eigenvalues of the corresponding bifurcation equation.…”